Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| 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 | |
975 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 975 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);(3,4,8,12)(5,13,16,32)(6,18,21,28)(7,22,23,11)(9,24,26,15)(10,19,30,34)(14,31,17,29)(20,27,33,25);(6,19)(7,23)(10,28)(11,22)(13,31)(14,17)(15,33)(18,30)(20,24)(21,34)(25,27)(29,32);(3,5)(4,9)(6,20)(7,14)(8,16)(10,29)(11,25)(12,26)(13,18)(15,34)(17,23)(19,24)(21,33)(22,27)(28,32)(30,31)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153629, C2×#153627, C2×#489349, … |
| maximal normal subgroup of | C2×#153629, C2×#153627, C2×#489349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396071);