Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13056 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1910 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,10,18,13,19,11,16)(9,20,23,14,24,31,21,32)(12,17,26,15,28,33,25,34)(22,35,36,38,27,29,37,30);(3,4,5,6)(7,9)(8,14)(10,21)(11,23)(12,27)(13,24)(15,29)(16,31)(17,30)(18,20)(19,32)(22,28)(25,37)(26,36)(33,38)(34,35);(7,10,13,11)(8,15,19,34)(9,21,24,23)(12,26,28,25)(14,29,32,35)(16,33,18,17)(20,30,31,38)(22,37,27,36)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266752, #72266755, C2×#499865, … |
| maximal normal subgroup of | #72266752, #72266755, C2×#499865, … |
| maximal quotient of | C2×#147497, #72266854, #72266882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401808);