Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37682 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(17,18)(19,28)(20,32)(21,26)(22,23)(24,30)(25,35)(27,37)(29,45)(31,40)(33,44)(34,43)(36,42)(38,48)(39,46)(41,47);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(18,23)(24,38)(25,32)(26,27)(28,40)(39,41)(42,43)(44,45);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,31,37,48)(18,24,27,28,23,38,26,40)(20,33,36,46,35,29,34,47)(25,41,42,45,32,39,43,44);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,20,22,35)(18,25,23,32)(19,29,31,33)(21,36,37,34)(24,39,38,41)(26,43,27,42)(28,44,40,45)(30,47,48,46);(2,6)(7,8)(10,14)(13,15)(17,21,22,37)(18,26,23,27)(19,30,31,48)(20,34,35,36)(24,40,38,28)(25,42,32,43)(29,46,33,47)(39,44,41,45)Polycyclic presentation:
2^9|b^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6818610);