Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12760 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2820 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 2820 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,13)(10,14,18,12)(11,15)(16,21,19,20)(17,22);(8,12)(10,18)(13,20)(14,15)(16,19)(21,22);(4,6)(7,9,11,17)(8,13,15,22)(10,19,18,16)(12,20,14,21)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153190, C2×#152603, C2×#511774, … |
| maximal normal subgroup of | C2×#153190, C2×#152603, C2×#511774, … |
| maximal quotient of | #72279556, C2×#133528, #72303854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400772);