Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13709 |
| automorphism | Aut(G) = #72319575 (order 2048) · Inn(G) = #2226 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2226 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,14)(3,13,18,45)(4,19)(5,24,26,11)(6,27,30,53)(7,31,16,9)(10,35)(12,41,43,29)(15,46)(17,50,52,40)(20,57)(21,39)(22,54)(23,58)(25,34,61,62)(28,44)(32,56)(33,60)(36,59)(37,63)(38,51)(42,47)(48,55)(49,64);(1,3,5,16,8,18,26,7)(2,9,11,13,14,31,24,45)(4,15,20,47,23,49,22,32)(6,17,25,41,30,52,61,29)(10,33,36,44,39,37,38,55)(12,34,40,27,43,62,50,53)(19,42,54,46,58,56,57,64)(21,48,59,63,35,28,51,60);(1,4)(2,10)(3,15)(5,20)(6,28)(7,32)(8,23)(9,33)(11,36)(12,42)(13,44)(14,39)(16,47)(17,51)(18,49)(19,53)(21,41)(22,26)(24,38)(25,60)(27,58)(29,35)(30,48)(31,37)(34,54)(40,46)(43,56)(45,55)(50,64)(52,59)(57,62)(61,63)Polycyclic presentation:
2^8|a^2=dg,b^2=d,d^2=h,e^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,e^c=efh,f^c=fg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151578, #151579, #151580, … |
| maximal normal subgroup of | #151578, #151579, #151580, … |
| maximal quotient of | #152281, #152448, #152802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6697);