Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | Aut(G) = #72287443 (order 4096) · Inn(G) = C2×D32 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C23 |
| prime radical | #2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
63 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)(3,15)(4,21)(5,28)(6,33)(7,37)(8,10)(9,16)(11,24)(12,23)(13,30)(14,35)(17,61)(18,65)(19,68)(20,54)(22,49)(25,81)(26,85)(27,53)(29,46)(31,93)(32,87)(34,55)(36,95)(38,56)(39,60)(40,73)(41,72)(42,98)(43,57)(44,62)(45,66)(47,107)(48,109)(50,113)(51,58)(52,59)(63,124)(64,123)(67,125)(69,76)(70,79)(71,83)(74,114)(75,104)(77,89)(78,88)(80,97)(82,111)(84,118)(86,112)(90,108)(91,110)(92,102)(94,106)(96,117)(99,127)(100,126)(101,128)(103,121)(105,122)(115,120)(116,119);(1,3,8,20)(2,9,10,39)(4,22,27,75)(5,29,32,92)(6,34,36,96)(7,38,35,19)(11,46,43,102)(12,49,44,104)(13,51,45,105)(14,52,37,42)(15,53,54,21)(16,57,60,24)(17,62,64,23)(18,66,67,30)(25,82,84,128)(26,86,83,74)(28,40,87,99)(31,94,77,91)(33,41,95,100)(47,108,103,124)(48,110,97,106)(50,114,88,112)(55,118,117,81)(56,71,68,85)(58,121,122,107)(59,80,98,109)(61,115,123,69)(63,78,90,113)(65,116,125,70)(72,120,126,76)(73,119,127,79)(89,111,93,101);(2,10)(9,39)(11,43)(12,44)(13,45)(14,37)(15,54)(21,53)(28,87)(33,95)(40,99)(41,100)(42,52)(46,102)(47,103)(48,97)(49,104)(50,88)(51,105)(55,117)(56,68)(61,123)(65,125)(69,115)(70,116)(71,85)(81,118)(89,93)(101,111)(106,110)(108,124)(112,114)Polycyclic presentation:
2^8|b^2=h,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #153597, #153608, #153611, … |
| maximal normal subgroup of | #153597, #153608, #153611, … |
| maximal quotient of | #153556, #153558, #153574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6731);