Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 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)(15,21)(16,22)(17,32)(18,24)(19,23)(20,25)(26,36)(27,38)(28,39)(29,40)(30,42)(31,43)(33,44)(34,37)(35,41);(5,7,8,12)(6,9,10,11)(13,15,20,28)(14,21,25,39)(16,29,31,33)(17,27,19,34)(18,30,26,35)(22,40,43,44)(23,37,32,38)(24,42,36,41);(1,2,3,4)(13,16)(14,22)(15,26)(17,30)(18,28)(19,35)(20,31)(21,36)(23,41)(24,39)(25,43)(27,33)(29,34)(32,42)(37,40)(38,44);(6,10)(9,11)(13,17,20,19)(14,23,25,32)(15,27,28,34)(16,30,31,35)(18,29,26,33)(21,37,39,38)(22,41,43,42)(24,44,36,40);(7,12)(9,11)(13,18,20,26)(14,24,25,36)(15,16,28,31)(17,33,19,29)(21,22,39,43)(23,40,32,44)(27,35,34,30)(37,42,38,41)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590523);