Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | #3006 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,19)(21,23)(22,24);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,21,22)(16,23,24,20);(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24);(1,3,4,7)(2,6,5,8);(9,11,14,13)(10,17,18,15)(12,19,21,22)(16,20,24,23)Polycyclic presentation:
2^8|b^2=h,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #114430, #114695, #114834, … |
| maximal normal subgroup of | #114430, #114695, #114834, … |
| maximal quotient of | #6370981, #6371788, #6371789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53551);