Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3200 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 3200 conjugacy classes of subgroups — too many to draw. Only its 258 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 — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(12,19)(14,20)(18,21)(22,24);(10,14,16,20)(11,17)(15,22,23,24)(18,21);(2,4)(9,10)(11,15)(12,20)(13,16)(14,19)(17,23)(18,22)(21,24);(6,8)(9,11,13,17)(10,15,16,23)(12,21,19,18)(14,22,20,24)Polycyclic presentation:
2^9|b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318059, #72318241, #72318621 |
| maximal normal subgroup of | #72318059, #72318241, #72318621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325010);