Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18)(19,28)(20,33)(21,27)(22,26)(23,37)(24,30)(25,35)(29,44)(31,43)(32,39)(34,40)(36,41)(38,47)(42,45)(46,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,38)(26,28)(27,39)(33,48)(34,46)(35,45)(36,47)(37,43)(40,44)(41,42);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,23,35,22,34,21,36)(18,25,27,33,26,41,37,40)(19,29,32,45,30,46,31,47)(24,38,39,48,28,42,43,44);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(18,26)(19,30)(20,34)(27,37)(31,32)(35,36)(38,42)(44,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,26,37)(19,31,30,32)(20,35,34,36)(24,39,28,43)(25,40,41,33)(29,45,46,47)(38,44,42,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=fi,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=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822626);