Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11533 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 226 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 — 4 generators on 26 points:
(1,2);(3,4,6,8,7,9,5,10)(11,12)(13,19,20,26)(14,22,24,17)(15,18)(16,21)(23,25);(11,13)(12,16)(14,23)(15,20)(17,26)(18,25)(19,22)(21,24);(3,5,7,6)(4,8,9,10)(13,20)(16,25)(19,26)(21,23)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#480400, #72240573, C2×#133582, … |
| maximal normal subgroup of | C2×#480400, #72240573, C2×#133582, … |
| maximal quotient of | #72261809, #72293125 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387535);