Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #30512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 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)(3,6)(4,8)(5,7)(17,18,22,29,23,28,39,27)(19,30,34,45,33,24,32,42)(20,35,37,43,21,26,36,25)(31,44,47,40,38,41,48,46);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,28,30)(20,31,21,38)(22,34,39,32)(25,40,43,46)(26,44,35,41)(27,45,29,42)(36,48,37,47);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,31)(21,23)(22,36)(24,40)(26,29)(27,35)(28,43)(30,46)(32,47)(33,38)(34,48)(37,39)(41,45)(42,44);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,21,23,20)(18,26,28,35)(19,31,33,38)(22,36,39,37)(24,41,30,44)(25,27,43,29)(32,48,34,47)(40,42,46,45);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,22,23,39)(18,27,28,29)(19,32,33,34)(20,36,21,37)(24,42,30,45)(25,35,43,26)(31,47,38,48)(40,44,46,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815033);