Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C2×#7734 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
2717 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 2717 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(15,18)(19,20)(21,23)(22,25)(24,26);(4,7)(5,10)(8,9)(11,12)(13,15)(14,16)(17,18);(11,13)(12,15)(14,17)(16,18)(20,23)(21,26)(24,25)74 conjugacy classes — view the full character table.
| maximal subgroup of | #72217679, C2×#177861, #72279350, … |
| maximal normal subgroup of | #72217679, C2×#177861, #72279350, … |
| maximal quotient of | C2×#102905, C2×#125647, C2×#125651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 72873);