Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12430 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 634 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);(3,4)(5,7)(6,9)(8,10)(12,16)(13,17)(14,20)(18,26)(19,24)(21,25)(22,23);(11,12,17,23,14,18,21,24,15,19,25,26,20,22,13,16);(4,7)(5,10)(8,9)98 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D8×D16, #72226895, #72226899 |
| maximal normal subgroup of | C2×D8×D16, #72226895, #72226899 |
| maximal quotient of | #72225569, C2×#123363 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397660);