Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18,20,24)(19,23,27,22)(21,25,26,30)(28,32,29,31);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,26,29,20,27,21,28)(18,22,30,32,24,23,25,31);(10,14)(13,15)(17,19,20,27)(18,23,24,22)(21,29,26,28)(25,31,30,32);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,24)(22,23)(25,30)(31,32)Polycyclic presentation:
2^8|a^2=e,b^2=fgh,c^2=h,d^2=efh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29078, #6460576, #6460605, … |
| maximal normal subgroup of | C2×#29078, #6460576, #6460605, … |
| maximal quotient of | #362579, #363683, #367864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22066);