Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C29 |
| direct product | C2×#12679 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | C22×C8 |
| prime radical | C26 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4378 conjugacy classes of subgroups; only the 690 normal subgroups are shown.
This group has 4378 conjugacy classes of subgroups — too many to draw. Only its 690 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,6)(7,8)(9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(4,6);(8,10)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#511741, #72286638, #72286655, … |
| maximal normal subgroup of | C2×#511741, #72286638, #72286655, … |
| maximal quotient of | #72287514 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400527);