Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#12997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,12,19,11,18,27,15,14,21,10,17,13,20,28,31)(9,22,25,30,29,16,32,34,26,36,23,37,24,35,38,33);(3,4,5,6)(8,15)(10,27)(11,13)(12,28)(16,30)(17,20)(18,19)(21,31)(22,34)(23,32)(24,29)(25,38)(33,36)(35,37);(7,9)(8,16)(10,25)(11,29)(12,23)(13,24)(14,26)(15,30)(17,34)(18,36)(19,33)(20,22)(21,35)(27,38)(28,32)(31,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401649);