Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2010 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)(15,20)(16,27)(17,22)(18,21)(19,23)(24,32)(25,34)(26,35)(28,37)(29,36)(30,38)(31,33)(39,42)(40,43)(41,44);(5,7,8,12)(6,9,10,11)(15,24)(16,18)(17,28)(20,32)(21,27)(22,37)(26,31)(29,30)(33,35)(36,38)(39,40)(42,43);(1,2,3,4)(13,15)(14,20)(16,24)(17,25)(18,31)(19,26)(21,33)(22,34)(23,35)(27,32)(28,39)(29,40)(30,41)(36,43)(37,42)(38,44);(6,10)(9,11)(13,16,19,18)(14,21,23,27)(15,24,26,31)(17,29,30,28)(20,33,35,32)(22,37,38,36)(25,40,41,39)(34,42,44,43);(7,12)(9,11)(13,17,19,30)(14,22,23,38)(15,25,26,41)(16,28,18,29)(20,34,35,44)(21,36,27,37)(24,39,31,40)(32,42,33,43)Polycyclic presentation:
2^9|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,d^b=di,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, 6590579);