Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #36015 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1684 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)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,20)(19,27,31,24)(21,25,37,33)(23,26,39,34)(28,32,41,30)(29,43,45,40)(35,38,42,36)(44,47,48,46);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,24,20,27)(21,29,37,45)(23,30,39,32)(25,40,33,43)(26,41,34,28)(35,48,42,44)(36,46,38,47);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,22,18)(19,28,31,41)(21,35,37,42)(23,26,39,34)(24,30,27,32)(25,38,33,36)(29,40,45,43)(44,47,48,46);(9,12,13,11)(10,16,15,14)(17,21)(18,25)(19,29)(20,33)(22,37)(23,36)(24,40)(26,35)(27,43)(28,44)(30,46)(31,45)(32,47)(34,42)(38,39)(41,48);(1,4,5,3)(2,6,8,7)(19,30)(21,36)(23,39)(24,41)(25,35)(26,34)(27,28)(31,32)(33,42)(37,38)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497607);