Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#12517 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 846 conjugacy classes of subgroups — too many to draw. Only its 250 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)(12,16)(13,20)(14,21)(17,22)(18,26)(19,25)(23,24);(11,12,17,26,14,18,13,16,15,19,20,24,21,23,22,25);(3,4,8,10,6,9,5,7)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#491469, D3×#12517, C2×#123412, … |
| maximal normal subgroup of | C2×#491469, C2×#123412, C2×#123419, … |
| maximal quotient of | #72225204, #72225205, D3×#12517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398659);