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)(15,21)(16,22)(17,23)(18,20)(19,24)(25,35)(26,36)(27,32)(28,37)(29,38)(30,33)(31,34)(39,44)(40,42)(41,43);(5,7,8,12)(6,9,10,11)(14,20)(15,25)(16,19)(17,29)(21,34)(23,37)(26,40)(27,31)(28,30)(32,35)(33,38)(43,44);(1,2,3,4)(13,15)(14,21)(16,25)(17,26)(18,27)(19,31)(20,32)(22,35)(23,36)(24,34)(28,39)(29,41)(30,40)(33,42)(37,44)(38,43);(6,10)(9,11)(13,16,18,19)(14,22,20,24)(15,25,27,31)(17,29,30,28)(21,35,32,34)(23,38,33,37)(26,41,40,39)(36,43,42,44);(7,12)(9,11)(13,17,18,30)(14,23,20,33)(15,26,27,40)(16,28,19,29)(21,36,32,42)(22,37,24,38)(25,39,31,41)(34,43,35,44)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,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, 6590587);