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 | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1540 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)(17,18,21,26,22,27,34,24)(19,23,29,37,30,38,45,35)(20,31,33,39,32,25,40,41)(28,42,44,46,43,36,47,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,38)(20,28,32,43)(21,29,34,45)(24,35,26,37)(25,36,31,42)(33,44,40,47)(39,46,41,48);(9,11,12,16)(10,13,14,15)(18,24)(21,34)(23,35)(25,39)(26,27)(29,45)(31,41)(33,40)(36,46)(37,38)(42,48)(44,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,39)(26,41)(27,31)(29,44)(30,43)(34,40)(35,46)(37,48)(38,42)(45,47);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(20,32)(25,31)(28,43)(33,40)(36,42)(39,41)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,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, 6980435);