Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,21,27,22,28)(19,29,35,39,31,24,32,37)(20,26,36,40,33,41,34,42)(30,43,46,48,44,38,45,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,31)(18,24,27,29)(20,30,33,44)(22,32,23,35)(25,37,28,39)(26,38,41,43)(34,45,36,46)(40,47,42,48);(9,11,12,16)(10,13,14,15)(18,25)(22,23)(24,37)(26,40)(27,28)(29,39)(32,35)(34,36)(38,47)(41,42)(43,48)(45,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,30)(21,33)(22,34)(23,36)(24,38)(25,40)(27,41)(28,42)(29,43)(31,44)(32,45)(35,46)(37,47)(39,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(19,31)(20,33)(24,29)(26,41)(32,35)(34,36)(37,39)(40,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6980374);