Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | #3150 | C4 |
| 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 | |
380 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 380 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:
(2,6,7,8)(9,10,14,19,15,20,23,24,17,21,13,22,16,11,12,18);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,24)(16,21)(17,20)(22,23);(1,3,5,4)(2,6,7,8)(9,12,15,14,17,23,16,13)(10,18,20,19,21,24,11,22)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213081, C2×#124917, #72217802, … |
| maximal normal subgroup of | C2×#124917, #72217802, #72217807, … |
| maximal quotient of | #72213037, #72213041, C2×#124917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32113);