Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18,23,27)(19,24,22,36)(20,30,33,43)(21,26,34,40)(25,28,39,32)(29,37,35,46)(31,42,44,48)(38,41,47,45);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,29,34,35)(25,30,39,43)(26,37,40,46)(31,41,44,45)(38,42,47,48);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,19,28,23,33,22,32)(18,25,36,43,27,39,24,30)(21,31,29,41,34,44,35,45)(26,38,46,48,40,47,37,42);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,21,23,34)(18,26,27,40)(19,29,22,35)(20,31,33,44)(24,37,36,46)(25,38,39,47)(28,41,32,45)(30,42,43,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(21,34)(26,40)(29,35)(31,44)(37,46)(38,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822659);