Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,29)(20,25)(21,34)(22,27)(23,28)(24,32)(26,36)(30,38)(31,39)(33,42)(35,41)(37,40)(43,44)(45,48)(46,47);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19)(18,24)(20,31)(21,30)(22,32)(23,37)(25,40)(26,38)(27,29)(28,39)(33,45)(34,43)(35,46)(36,44)(41,48)(42,47);(2,6)(5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(21,35)(22,23)(26,42)(27,28)(30,44)(31,37)(33,36)(34,41)(38,43)(39,40);(10,13)(14,16)(17,21,22,36)(18,26,27,34)(19,30,32,44)(20,33,23,35)(24,38,29,43)(25,41,28,42)(31,45,37,46)(39,47,40,48);(18,27)(19,31)(20,23)(24,39)(26,34)(29,40)(30,45)(32,37)(33,35)(38,47)(43,48)(44,46)Polycyclic presentation:
2^9|b^2=g,c^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585990);