Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,29)(23,31,24,27)(26,35,34,30)(28,32,36,33);(1,2,3,4)(6,9)(7,11)(13,15,16,19)(14,17,18,20)(21,23,25,24)(22,27,29,31)(26,32,34,33)(28,30,36,35);(2,4)(5,7)(6,10)(8,11)(9,12)(21,24,25,23)(22,28,29,36)(26,32,34,33)(27,30,31,35);(7,11)(10,12)(14,18)(17,20)(23,24)(27,31)(28,36)(32,33)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378685);