Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10)(11,15)(12,23)(13,17)(14,19)(16,22)(18,20)(21,24);(5,6,8,7)(9,11,20,24,14,22,13,12)(10,15,18,21,19,16,17,23);(2,4)(6,7)(9,10)(11,21)(12,16)(13,18)(14,19)(15,24)(17,20)(22,23);(5,7,8,6)(9,12,13,22,14,24,20,11)(10,16,18,23,19,15,17,21)Polycyclic presentation:
2^9|b^2=gi,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215350, #72216163, #72217565 |
| maximal normal subgroup of | #72215350, #72216163, #72217565 |
| maximal quotient of | #72215281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382936);