Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C23×#454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
22658 conjugacy classes of subgroups; only the 4002 normal subgroups are shown.
This group has 22658 conjugacy classes of subgroups — too many to draw. Only its 4002 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) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4);(5,6);(8,12)(10,17)(13,21)(14,18)(15,19)(20,22);(7,8)(9,13)(10,18)(11,14)(12,17)(15,20)(16,22)(19,21);(7,9,11,16)(8,13,14,22)(10,19,17,15)(12,20,18,21)128 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#3021, C2×#499867, #72266734, … |
| maximal normal subgroup of | C23×#3021, C2×#499867, #72266734, … |
| maximal quotient of | C22×#32889, C22×#18186, C2×#6346730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482005);