Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#2833 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4);(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)(21,22,26,29)(23,30,24,27)(25,28,33,36)(31,35,32,34);(5,7)(6,9)(8,11)(10,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,24)(22,27,29,30)(25,32,33,31)(28,35,36,34);(6,10)(9,12)(15,19)(17,20)(21,24,26,23)(22,27,29,30)(25,31,33,32)(28,35,36,34);(7,11)(9,12)(21,25,26,33)(22,28,29,36)(23,31,24,32)(27,34,30,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451551);