Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 56 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23)(25,26,28,32,29,33,38,44,30,34,39,45,40,46,50,54)(27,31,36,42,37,43,49,53,35,41,47,51,48,52,55,56);(1,3)(2,5)(4,8)(6,7)(9,11,13,19)(10,14,15,22)(12,20,21,18)(16,24,23,17)(25,27)(26,31)(28,36)(29,37)(30,35)(32,42)(33,43)(34,41)(38,49)(39,47)(40,48)(44,53)(45,51)(46,52)(50,55)(54,56);(10,15)(14,22)(16,23)(17,24)(27,35)(31,41)(36,47)(37,48)(42,51)(43,52)(49,55)(53,56)Polycyclic presentation:
2^9|a^2=e,b^2=g,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,c^b=ci,d^b=dg,e^b=efg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31741);