Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3506 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,10,23)(3,13)(5,19)(9,22,26,25)(11,24,12,20)(14,29)(15,16)(27,32,28,31);(1,2)(3,8)(4,9)(5,20)(6,11)(7,12)(10,21)(13,24)(14,22)(15,25)(16,31)(17,27)(18,28)(19,23)(26,30)(29,32);(1,3,7,5)(2,8,12,20)(4,15,18,29)(6,13,21,19)(9,25,28,32)(10,23,11,24)(14,17,16,30)(22,27,31,26);(1,4,7,18)(2,9,12,28)(3,14,5,16)(6,17,21,30)(8,22,20,31)(10,26,11,27)(13,29,19,15)(23,25,24,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh49 conjugacy classes — view the full character table.
| maximal subgroup of | #253512, #253518, #253524, … |
| maximal normal subgroup of | #253512, #253518, #253524, … |
| maximal quotient of | #160656, #160657, #160664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10617);