Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,20,26)(15,21,29,37)(16,22,27,35)(17,23,19,25)(18,24,34,42)(28,36,43,44)(30,38,32,40)(31,39,33,41);(5,7)(6,9)(8,12)(10,11)(13,15,20,29)(14,21,26,37)(16,17,27,19)(18,32,34,30)(22,23,35,25)(24,40,42,38)(28,31,43,33)(36,39,44,41);(1,2,3,4)(13,16,19,15,20,27,17,29)(14,22,25,21,26,35,23,37)(18,32,33,43,34,30,31,28)(24,40,41,44,42,38,39,36);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,27,29,16)(18,31,34,33)(21,35,37,22)(24,39,42,41)(28,32,43,30)(36,40,44,38);(7,12)(9,11)(13,18,19,33,20,34,17,31)(14,24,25,41,26,42,23,39)(15,28,27,32,29,43,16,30)(21,36,35,40,37,44,22,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722125);