Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| 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, prime radical | C24 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(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);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(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);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(27,35)(31,41)(36,47)(37,48)(42,51)(43,52)(49,55)(53,56)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,f^c=fg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42382);