Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,31,40)(20,25,34,44)(21,35,22,26)(28,37,30,39)(29,38,46,47)(32,41,45,48)(33,43,36,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,32)(21,30)(22,28)(23,31)(25,41)(26,37)(27,40)(29,33)(34,45)(35,39)(36,46)(38,43)(42,47)(44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,28)(21,22)(24,37)(26,35)(29,32)(30,31)(33,36)(38,41)(39,40)(42,43)(45,46)(47,48);(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,38)(26,42)(27,44)(28,32)(30,45)(31,46)(35,43)(37,41)(39,48)(40,47);(17,21,23,22)(18,26,27,35)(19,30,31,28)(20,33,34,36)(24,37,40,39)(25,42,44,43)(29,45,46,32)(38,41,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348211);