Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10,14,17,15,11,13,19)(12,16,21,23,22,18,20,24);(5,6,7,8)(9,11,15,10)(12,18,22,16)(13,17,14,19)(20,23,21,24);(1,2,3,4)(9,12,14,21,15,22,13,20)(10,16,17,23,11,18,19,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10934, #354098, #354099, … |
| maximal normal subgroup of | C2×#10934, #354098, #354099, … |
| maximal quotient of | #110163, #110164, #110165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3922);