Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C23 | C2×#104 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(10,15,16,21,17,23,22,24)(11,18)(12,20);(1,3,6,2)(4,8,5,7)(9,10,13,17)(11,15,18,23)(12,21,20,24)(14,22,19,16);(1,4,6,5)(2,7,3,8)(9,11,19,12,13,18,14,20)(10,15,16,21,17,23,22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fgh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cfh,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213082, #72217810, #72217815 |
| maximal normal subgroup of | #72217810, #72217815 |
| maximal quotient of | #72213038, #72213041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32114);