Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1587 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,31)(21,36)(22,38)(23,28)(25,35)(26,30)(27,39)(29,45)(32,43)(33,40)(34,47)(37,41)(42,46)(44,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,28,26)(20,32,34,33)(21,22,37,39)(25,42,44,45)(27,36,38,41)(29,35,46,48)(31,43,47,40);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,35,23,34,39,48)(18,25,27,31,28,44,38,47)(19,29,21,33,30,46,37,32)(24,40,41,45,26,43,36,42);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,30,23,37,39,19)(18,26,27,36,28,24,38,41)(20,33,35,46,34,32,48,29)(25,43,31,42,44,40,47,45);(2,6)(7,8)(10,14)(13,15)(20,34)(25,44)(29,46)(31,47)(32,33)(35,48)(40,43)(42,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fg,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499718);