Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1385 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)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,35)(21,39)(22,28)(23,29)(24,27)(25,34)(26,38)(31,42)(32,43)(33,47)(36,46)(37,44)(40,48)(41,45);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,36)(21,32)(22,38)(23,34)(24,41)(26,46)(27,43)(28,35)(29,30)(31,37)(33,40)(39,45)(42,47)(44,48);(9,12,13,11)(10,14,15,16)(17,20,24,37,23,38,21,33)(18,26,27,47,29,35,39,44)(19,31,32,36,34,40,41,22)(25,42,45,46,30,48,43,28);(2,6)(5,7)(17,21,23,24)(18,27,29,39)(19,32,34,41)(20,37,38,33)(22,40,36,31)(25,43,30,45)(26,44,35,47)(28,42,46,48);(3,8)(5,7)(17,22,24,31,23,36,21,40)(18,28,27,42,29,46,39,48)(19,33,32,20,34,37,41,38)(25,44,45,26,30,47,43,35)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727451);