Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3291 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 931 conjugacy classes of subgroups — too many to draw. Only its 154 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)(5,14)(6,17,19,34)(7,12)(8,22)(9,24)(10,16)(11,15,28,33)(13,21)(18,26,25,31)(20,29)(23,30,32,27);(3,5,7,16)(4,10,12,14)(6,15,19,33)(8,9,20,21)(11,26,28,31)(13,29,24,22)(17,27,34,30)(18,23,25,32);(3,6)(4,11)(5,15)(7,19)(8,23)(9,25)(10,26)(12,28)(13,30)(14,31)(16,33)(17,29)(18,21)(20,32)(22,34)(24,27)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153641, C2×#153639, C2×#489349, … |
| maximal normal subgroup of | C2×#153641, C2×#153639, C2×#489349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396122);