Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| 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 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,2,4,5)(3,7,8,6);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,3,4,8)(2,6,5,7);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|b^2=f,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,g^c=gh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #61303⋊C3 |
| maximal subgroup of | #221019, #221031, #221042, … |
| maximal normal subgroup of | #221019, #221031, #221042, … |
| maximal quotient of | Q8×#416, #6379446, #6380325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54291);