Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7107 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #331 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,13,19,15,17,21,24,16,20,23,26,14,18,22,25);(3,5)(4,7)(6,10)(8,9)(12,17)(13,21)(14,15)(18,20)(19,26)(22,23)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213160, C22×#7107, C2×#125647, … |
| maximal normal subgroup of | #72213160, C22×#7107, C2×#125647, … |
| maximal quotient of | C2×#92827, #72213553, C22×#7107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31252);