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 |
| 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 | |
67 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)(3,8)(4,19)(5,23)(6,26)(7,9)(10,14)(11,15)(12,20)(13,24)(16,37)(17,38)(18,27)(21,54)(22,50)(25,56)(28,40)(29,39)(30,47)(31,42)(32,45)(33,53)(34,46)(35,49)(36,55)(41,57)(43,52)(44,51)(48,61)(58,63)(59,62)(60,64);(2,8)(3,14)(4,15)(5,20)(6,24)(9,27)(10,11)(12,37)(13,38)(16,46)(17,49)(18,42)(19,28)(21,55)(22,45)(23,29)(25,53)(26,30)(31,32)(33,61)(34,52)(35,51)(36,57)(39,40)(41,47)(43,44)(48,64)(50,58)(54,60)(56,59)(62,63);(2,9)(8,27)(10,31)(11,32)(12,33)(13,26)(19,50)(23,56)(28,58)(29,59)(30,38)(34,60)(35,57)(36,51)(37,61)(52,54)Polycyclic presentation:
2^8|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 | #153598, #153607, #153609, … |
| maximal normal subgroup of | #153598, #153607, #153609, … |
| maximal quotient of | #153556, #153557, #153574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6730);