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 | |
1993 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)(13,14,19,25)(15,21,27,39)(16,28,31,22)(17,32,20,23)(18,24,34,37)(26,40,35,38)(29,44,33,41)(30,42,36,43);(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,29)(17,26)(18,30)(19,27)(20,35)(22,41)(23,38)(24,42)(25,39)(28,44)(31,33)(32,40)(34,36)(37,43);(1,2,3,4)(13,16,19,31)(14,22,25,28)(15,18,27,34)(17,30,20,36)(21,37,39,24)(23,42,32,43)(26,33,35,29)(38,41,40,44);(6,10)(9,11)(13,17,19,20)(14,23,25,32)(15,26,27,35)(16,30,31,36)(18,33,34,29)(21,38,39,40)(22,42,28,43)(24,44,37,41);(7,12)(9,11)(13,18,19,34)(14,24,25,37)(15,16,27,31)(17,29,20,33)(21,28,39,22)(23,41,32,44)(26,36,35,30)(38,42,40,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,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, 6590531);