Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2203 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6)(7,16)(8,20)(9,12)(10,15)(11,17)(13,23)(14,25)(18,33)(19,31)(21,29)(22,30)(24,34)(26,32)(27,35)(28,36);(5,7,10,19)(6,11,15,29)(8,21,23,17)(9,24,25,18)(12,27,14,28)(13,31,20,16)(22,35,26,36)(30,33,32,34);(6,12)(9,25)(11,27)(13,32)(14,15)(16,33)(18,24)(20,30)(22,26)(28,29)(31,34)(35,36)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402954);