Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12300 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| prime radical | C2×#3197 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 138 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);(3,4,8,23,9,13,6,11)(5,14,15,28,17,32,24,31)(7,19,22,12,21,29,18,30)(10,20,25,16,27,33,26,34);(3,5)(4,10)(6,15)(7,20)(8,24)(9,17)(11,25)(12,28)(13,27)(14,29)(16,18)(19,32)(21,33)(22,34)(23,26)(30,31);(3,6,9,8)(4,11,13,23)(5,15,17,24)(7,18,21,22)(10,25,27,26)(12,29,30,19)(14,31,32,28)(16,33,34,20)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486694, C2×#485587, C2×#486082, … |
| maximal normal subgroup of | C2×#486694, C2×#485587, C2×#486082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391764);