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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1389 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)(19,25)(20,35)(21,27)(22,28)(23,29)(24,30)(26,37)(31,47)(32,42)(33,43)(34,44)(36,46)(38,45)(39,41)(40,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,22)(21,32)(23,34)(24,40)(26,45)(27,42)(28,35)(29,44)(30,48)(31,36)(33,39)(37,38)(41,43)(46,47);(9,12,13,11)(10,14,15,16)(17,20,24,36,23,37,21,33)(18,26,30,43,29,35,27,46)(19,31,32,22,34,39,40,38)(25,41,42,45,44,47,48,28);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,32,34,40)(20,36,37,33)(22,31,38,39)(25,42,44,48)(26,43,35,46)(28,47,45,41);(3,8)(5,7)(17,22,21,31,23,38,24,39)(18,28,27,47,29,45,30,41)(19,33,40,37,34,36,32,20)(25,43,48,26,44,46,42,35)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=i,e^2=hi,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, 6727452);