Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 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)(3,6)(4,5)(7,8)(17,18,22,27)(19,23,30,39)(20,25,32,43)(21,34,36,26)(24,35,41,33)(28,42,29,38)(31,47,46,40)(37,45,48,44);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,39)(20,28,32,29)(21,31,36,46)(24,37,41,48)(25,42,43,38)(26,40,34,47)(33,44,35,45);(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,34)(26,43)(27,41)(29,30)(31,45)(33,36)(38,40)(39,48)(42,47)(44,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,41)(27,43)(28,30)(31,45)(35,36)(37,40)(39,42)(44,46)(47,48);(10,14)(13,15)(19,30)(20,32)(23,39)(24,41)(25,43)(31,46)(33,35)(40,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^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=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6408603);