Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1048 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)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,26,31)(23,32,27,28)(24,34,35,30)(25,36,33,29);(6,9)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23,26,27)(22,28,31,32)(24,25,35,33)(29,30,36,34);(5,7)(6,10)(8,12)(9,11)(15,19)(17,20)(21,24,27,25,26,35,23,33)(22,29,32,30,31,36,28,34);(2,4)(7,12)(10,11)(21,25,27,35,26,33,23,24)(22,30,32,36,31,34,28,29)Polycyclic presentation:
2^9|a^2=egi,b^2=i,c^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370584);