Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C23×#357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
5330 conjugacy classes of subgroups; only the 1602 normal subgroups are shown.
This group has 5330 conjugacy classes of subgroups — too many to draw. Only its 1602 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) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4);(5,6);(7,8,9,10)(11,12)(13,16)(14,23)(15,19)(17,20)(18,21)(22,25)(24,26);(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24)176 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#2177, C2×#498148, C2×D4×#357, … |
| maximal normal subgroup of | C23×#2177, C2×#498148, C2×D4×#357, … |
| maximal quotient of | C2×D4×#357, C24×#357, C23×#1419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529683);