Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22,24,28,25,29,32,27)(23,26,31,35,30,34,36,33);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15,16,19)(14,17,18,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,35)(29,34)(32,36);(1,2)(3,4)(7,11)(10,12)(15,19)(17,20)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(23,30)(26,34)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 381093);