Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×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 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22,27,33,25,24,26,29)(23,30,34,35,31,28,32,36);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,25,31)(22,28,24,30)(26,32,27,34)(29,35,33,36);(5,7)(6,10)(8,11)(9,12)(15,19)(17,20)(21,24,25,22)(23,28,31,30)(26,29,27,33)(32,36,34,35);(2,4)(7,11)(10,12)(22,24)(23,31)(29,33)(32,34)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=i,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, 374222);