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 | |
1297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,35,24)(19,28,31,37,30,23,36,39)(20,32,34,40,33,25,41,42)(29,38,45,48,44,43,46,47);(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,21,30)(18,23,26,28)(20,29,33,44)(22,31,35,36)(24,37,27,39)(25,38,32,43)(34,45,41,46)(40,47,42,48);(9,11,12,16)(10,13,14,15)(18,24)(22,35)(23,37)(25,40)(26,27)(28,39)(31,36)(32,42)(34,41)(38,47)(43,48)(45,46);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,38)(24,40)(26,32)(27,42)(28,43)(30,44)(31,45)(35,41)(36,46)(37,47)(39,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,30)(20,33)(23,28)(25,32)(31,36)(34,41)(37,39)(40,42)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6980118);