Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11915 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1533 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
716 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,12,7,13,20,31,8,10,19,27,22,11,28,29)(5,14,15,32,16,9,23,25,17,26,33,30,21,24,18,34);(3,5,8,17)(4,9,10,24)(6,18,19,23)(7,21,22,16)(11,26,13,14)(12,30,27,32)(15,20,33,28)(25,29,34,31);(4,10)(6,19)(9,24)(11,13)(14,26)(15,33)(18,23)(20,28)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#133262, C2×#143165, C2×#143163, … |
| maximal normal subgroup of | C2×#133262, C2×#143165, C2×#143163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390214);