Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,5)(6,8)(9,10,13,16)(11,14,12,15);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15);(10,15)(11,12)(14,16);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=df,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221071, #221081, #221091, … |
| maximal normal subgroup of | #221071, #221081, #221091, … |
| maximal quotient of | #6379542, #6379543, #6380376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54427);