Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1331 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)(10,14)(12,15)(17,18,22,27)(19,23,30,26)(20,31,33,36)(21,25,35,42)(24,38,40,28)(29,37,46,43)(32,47,34,48)(39,45,41,44);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,30)(18,23,27,26)(20,28,33,38)(21,29,35,46)(24,36,40,31)(25,37,42,43)(32,44,34,45)(39,48,41,47);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,15,14,12)(17,20,19,28,22,33,30,38)(18,24,26,31,27,40,23,36)(21,34,29,45,35,32,46,44)(25,41,43,48,42,39,37,47);(3,8)(6,7)(10,14)(11,16)(17,21,22,35)(18,25,27,42)(19,29,30,46)(20,32,33,34)(23,37,26,43)(24,39,40,41)(28,44,38,45)(31,47,36,48);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,16,14,11)(21,35)(25,42)(29,46)(32,34)(37,43)(39,41)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fg,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6955761);