Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(8,12,14,16)(9,13)(10,18)(15,20,22,19)(17,21);(7,8,13,20,18,22,9,16,11,14,21,19,10,15,17,12);(3,4,5,6)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#496124, #72224819, #72224821, … |
| maximal normal subgroup of | C2×#496124, #72224819, #72224821, … |
| maximal quotient of | #72225248, #72225252, C2×#141751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402328);