Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18)(19,28)(20,32)(21,26)(22,24)(23,30)(25,34)(27,36)(29,38)(31,39)(33,41)(35,42)(37,44)(40,46)(43,47)(45,48);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,33)(21,30)(22,31)(24,37)(25,41)(26,28)(27,39)(29,34)(32,38)(35,45)(36,44)(40,48)(42,47)(43,46);(9,11,12,16)(10,13,14,15)(18,24)(22,36)(23,37)(25,40)(26,27)(28,39)(31,44)(32,42)(35,46)(38,47)(41,48)(43,45);(1,3,4,7)(2,6,5,8)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,40)(26,32)(27,42)(28,41)(30,33)(31,43)(36,46)(37,47)(39,48)(44,45);(2,5)(6,8)(10,14)(13,15)(18,26)(19,30)(20,34)(24,27)(31,44)(35,46)(38,41)(47,48)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6661642);