Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#33567 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| socle | C22 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4005 conjugacy classes of subgroups; only the 570 normal subgroups are shown.
This group has 4005 conjugacy classes of subgroups — too many to draw. Only its 570 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,13)(6,11)(7,10)(8,17)(9,14)(12,15)(16,18);(4,10)(5,8)(7,15)(11,13)(12,14)(17,18);(3,5)(6,15)(7,16)(8,9)(10,12)(13,17);(3,6,9,16)(4,11,14,18)(5,15,8,7)(10,13,12,17)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#511838, C2×#511839, #72261055, … |
| maximal normal subgroup of | C2×#511838, C2×#511839, #72261055, … |
| maximal quotient of | #72261890, C2×#498989 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530093);