Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | Aut(G) = #72230814 (order 8192) · Inn(G) = D64 · Out(G) = C2×C32 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic64 |
| quaternion | Q256 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2 | D64 |
| commutator, Frattini, cosocle | C64 | C22 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ G | |
| LECS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 256 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,256,255,254,253,252,251,250,249,248,247,246,245,244,243,242,241,240,239,238,237,236,235,234,233,232,231,230,229,228,227,226,225,224,223,222,221,220,219,218,217,216,215,214,213,212,211,210,209,208,207,206,205,204,203,202,201,200,199,198,197,196,195,194,193,192,191,190,189,188,187,186,185,184,183,182,181,180,179,178,177,176,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130);(1,129,65,193)(2,130,66,194)(3,131,67,195)(4,132,68,196)(5,133,69,197)(6,134,70,198)(7,135,71,199)(8,136,72,200)(9,137,73,201)(10,138,74,202)(11,139,75,203)(12,140,76,204)(13,141,77,205)(14,142,78,206)(15,143,79,207)(16,144,80,208)(17,145,81,209)(18,146,82,210)(19,147,83,211)(20,148,84,212)(21,149,85,213)(22,150,86,214)(23,151,87,215)(24,152,88,216)(25,153,89,217)(26,154,90,218)(27,155,91,219)(28,156,92,220)(29,157,93,221)(30,158,94,222)(31,159,95,223)(32,160,96,224)(33,161,97,225)(34,162,98,226)(35,163,99,227)(36,164,100,228)(37,165,101,229)(38,166,102,230)(39,167,103,231)(40,168,104,232)(41,169,105,233)(42,170,106,234)(43,171,107,235)(44,172,108,236)(45,173,109,237)(46,174,110,238)(47,175,111,239)(48,176,112,240)(49,177,113,241)(50,178,114,242)(51,179,115,243)(52,180,116,244)(53,181,117,245)(54,182,118,246)(55,183,119,247)(56,184,120,248)(57,185,121,249)(58,186,122,250)(59,187,123,251)(60,188,124,252)(61,189,125,253)(62,190,126,254)(63,191,127,255)(64,192,128,256)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh67 conjugacy classes — view the full character table.
| maximal subgroup of | SD512, Q512, C2×Q256, … |
| maximal normal subgroup of | SD512, Q512, C2×Q256, … |
| maximal quotient of | #94833, #94840, C2×Q256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 541);