Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
690 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,8)(9,10)(11,12)(13,16)(14,23)(15,19)(17,20)(18,21)(22,25)(24,26);(3,4,8,10,6,9,5,7);(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24)86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216276, C2×#123275, #72225188, … |
| maximal normal subgroup of | #72216276, C2×#123275, #72225188, … |
| maximal quotient of | C2×#151634, #72225339, #72225448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397642);