Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1078 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18);(3,8)(5,7);(2,6)(5,7)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #249884, #249887, #249916, … |
| maximal normal subgroup of | #249884, #249887, #249916, … |
| maximal quotient of | #6386037, #6387626, #6387640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54612);