Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 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)(11,16)(12,14)(13,15)(17,18)(19,24)(20,25)(21,26)(22,36)(23,28)(27,30)(29,38)(31,39)(32,40)(33,41)(34,42)(35,44)(37,43)(45,47)(46,48);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,32)(21,22)(23,31)(25,40)(26,36)(27,41)(28,39)(29,34)(30,33)(35,37)(38,42)(43,44)(45,46)(47,48);(9,11,13,12)(10,14,15,16)(17,20,33,46,23,35,21,34)(18,25,41,48,28,44,26,42)(19,29,22,37,31,45,30,32)(24,38,36,43,39,47,27,40);(2,6)(5,7)(17,21,23,33)(18,26,28,41)(19,22,31,30)(20,34,35,46)(24,36,39,27)(25,42,44,48)(29,37,45,32)(38,43,47,40);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,22,33,31,23,30,21,19)(18,27,41,24,28,36,26,39)(20,32,46,29,35,37,34,45)(25,43,48,47,44,40,42,38)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6741592);