Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25143 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,17,13,18,11,16)(9,14,21,30,22,31,20,29)(10,23,25,33,26,34,27,15)(19,35,36,32,24,38,37,28);(4,6)(7,9,12,21,13,22,11,20)(8,14,17,30,18,31,16,29)(10,24,25,37,26,19,27,36)(15,32,23,38,33,28,34,35);(7,10,13,26)(8,15,18,33)(9,19,22,24)(11,25,12,27)(14,28,31,32)(16,23,17,34)(20,36,21,37)(29,35,30,38);(4,6)(7,11,13,12)(8,16,18,17)(9,20,22,21)(10,25,26,27)(14,29,31,30)(15,23,33,34)(19,36,24,37)(28,35,32,38)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346074);