Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 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)(3,8)(4,6)(5,7)(17,18,23,27)(19,24,30,39)(20,31,33,45)(21,26,22,36)(25,32,41,34)(28,42,44,47)(29,38,35,40)(37,46,48,43);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,40)(27,39)(31,42)(32,46)(33,44)(34,43)(36,38)(41,48)(45,47);(9,11)(10,13)(12,16)(14,15)(17,20,21,32,23,33,22,34)(18,25,36,45,27,41,26,31)(19,28,35,46,30,44,29,43)(24,37,38,47,39,48,40,42);(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,33,34)(24,38,39,40)(25,31,41,45)(28,43,44,46)(37,47,48,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,30)(20,33)(24,39)(25,41)(29,35)(31,45)(32,34)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415593);