Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C22×#2615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
27866 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.
This group has 27866 conjugacy classes of subgroups — too many to draw. Only its 4258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(10,12)(13,14)(15,18)(16,17)(19,20);(6,8)(10,12)(14,17)(18,19);(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,17)(18,20);(13,15,16,20)(14,18,17,19)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231659, C22×#35680, C2×#234827, … |
| maximal normal subgroup of | #72231659, C22×#35680, C2×#234827, … |
| maximal quotient of | C2×#1393171, C23×#2615, C2×#1394859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534958);