Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53169 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
13744 conjugacy classes of subgroups; only the 2946 normal subgroups are shown.
This group has 13744 conjugacy classes of subgroups — too many to draw. Only its 2946 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 — 6 generators on 26 points:
(1,2);(3,4,7,9)(5,6,10,8)(11,12,14,15)(13,17,18,16)(21,25)(24,26);(3,5,7,10)(4,8,9,6)(12,15)(16,17)(20,23)(24,26);(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26);(3,5)(4,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,25)(23,26);(3,6)(4,5)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,23)(25,26)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8108057, C2×#8797764, C2×#9223101, … |
| maximal normal subgroup of | C2×#8108057, C2×#8797764, C2×#9223101, … |
| maximal quotient of | C2×#2032030, C22×#53169, C2×#2032059 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012853);