Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = C2×#72224662 (order 4096) · Inn(G) = D32 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D32 |
| commutator, Frattini, cosocle | C32 | C23 |
| socle | C2 | C2×D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,16)(4,11)(5,25)(6,32)(7,37)(8,40)(9,15)(10,18)(12,28)(13,27)(14,34)(17,56)(19,64)(20,66)(21,59)(22,68)(23,73)(24,55)(26,52)(29,87)(30,90)(31,77)(33,49)(35,96)(36,93)(38,57)(39,98)(41,58)(42,60)(43,79)(44,78)(45,101)(46,63)(47,71)(48,70)(50,111)(51,86)(53,83)(54,62)(61,118)(65,120)(67,116)(69,108)(72,122)(74,107)(75,81)(76,84)(80,115)(82,92)(85,100)(88,102)(89,127)(91,114)(94,112)(95,110)(97,109)(99,117)(103,119)(104,128)(105,123)(106,125)(113,121)(124,126);(1,3)(2,10)(4,17)(5,26)(6,33)(7,38)(8,41)(9,21)(11,42)(12,49)(13,52)(14,54)(15,46)(16,25)(18,28)(19,27)(20,34)(22,69)(23,74)(24,67)(29,88)(30,91)(31,80)(32,43)(35,97)(36,95)(37,44)(39,99)(40,45)(47,107)(48,108)(50,112)(51,110)(53,115)(55,103)(56,68)(57,87)(58,90)(59,77)(60,71)(61,70)(62,111)(63,86)(64,75)(65,83)(66,76)(72,123)(73,102)(78,81)(79,84)(82,94)(85,101)(89,128)(92,114)(93,105)(96,104)(98,106)(100,109)(113,118)(116,122)(117,127)(119,121)(120,124)(125,126);(1,4,9,8)(2,11,15,40)(3,17,21,41)(5,22,31,30)(6,23,36,35)(7,24,39,34)(10,42,46,45)(12,47,51,100)(13,48,53,92)(14,37,55,98)(16,56,59,58)(18,60,63,101)(19,61,65,94)(20,38,67,99)(25,68,77,90)(26,69,80,91)(27,70,83,82)(28,71,86,85)(29,72,89,84)(32,73,93,96)(33,74,95,97)(43,102,105,104)(44,103,106,54)(49,107,110,109)(50,75,113,124)(52,108,115,114)(57,116,117,66)(62,78,119,125)(64,118,120,112)(76,87,122,127)(79,88,123,128)(81,121,126,111)Polycyclic presentation:
2^8|c^2=h,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #94834, #94835, #153595, … |
| maximal normal subgroup of | #94834, #94835, #153595, … |
| maximal quotient of | #153555, #153573, C4×D64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6729);