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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 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)(9,10)(11,15)(12,14)(13,16)(17,18,21,26)(19,28,31,23)(20,33,34,25)(22,37,36,27)(24,38,42,35)(29,43,30,40)(32,41,48,45)(39,46,44,47);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,21,31)(18,23,26,28)(20,29,34,30)(22,32,36,48)(24,39,42,44)(25,43,33,40)(27,41,37,45)(35,46,38,47);(11,16)(13,15)(17,20)(18,24)(19,29)(21,34)(22,38)(23,39)(25,27)(26,42)(28,44)(30,31)(32,47)(33,37)(35,36)(40,45)(41,43)(46,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,40)(24,27)(26,33)(28,43)(29,31)(32,47)(36,38)(37,42)(39,45)(41,44)(46,48);(2,5)(6,8)(19,31)(20,34)(23,28)(24,42)(25,33)(32,48)(35,38)(41,45)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6679154);