Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #32715 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1658 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)(19,24)(20,31)(21,26)(22,36)(23,27)(25,33)(28,42)(29,38)(30,39)(32,41)(34,45)(35,40)(37,43)(44,47)(46,48);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,27,39)(20,28,34,44)(21,35,22,29)(25,37,41,48)(26,40,36,38)(31,42,45,47)(32,46,33,43);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,23,34,21,32)(18,25,26,31,27,41,36,45)(19,28,29,43,30,44,35,46)(24,37,40,42,39,48,38,47);(2,6)(3,8)(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,34,33)(24,38,39,40)(25,31,41,45)(28,43,44,46)(37,47,48,42);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(19,30)(24,39)(28,44)(29,35)(37,48)(38,40)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=i,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=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819887);