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 | #29606 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(17,18,23,27)(19,24,22,37)(20,30,33,39)(21,26,36,44)(25,28,41,31)(29,38,34,42)(32,47,35,48)(40,45,43,46);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,34,36,29)(25,39,41,30)(26,42,44,38)(32,45,35,46)(40,47,43,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,31,23,33,19,28)(18,25,24,30,27,41,37,39)(21,35,34,45,36,32,29,46)(26,43,38,48,44,40,42,47);(2,6)(3,8)(11,16)(13,15)(17,21)(18,26)(19,29)(20,32)(22,34)(23,36)(24,38)(25,40)(27,44)(28,45)(30,47)(31,46)(33,35)(37,42)(39,48)(41,43);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(20,33)(21,36)(25,41)(26,44)(28,31)(29,34)(30,39)(38,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819997);