Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,18)(11,15,20,24)(12,17,22,23)(13,16,21,19);(1,3)(2,6)(4,7)(5,8)(9,11,14,20)(10,15,18,24)(12,21,22,13)(16,17,19,23);(1,4,5,2)(3,7,8,6)(10,16)(11,20)(13,21)(15,23)(17,24)(18,19);(3,8)(6,7)(9,12)(10,17)(11,21)(13,20)(14,22)(15,19)(16,24)(18,23);(10,18)(11,20)(12,22)(16,19)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130850, #130852, #250547, … |
| maximal normal subgroup of | #130850, #130852, #250547, … |
| maximal quotient of | #6379329, #6379861, #6380090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54281);