Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #33187 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1414 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)(19,24)(20,34)(21,26)(22,27)(23,28)(25,37)(29,46)(30,40)(31,41)(32,42)(33,43)(35,47)(36,44)(38,45)(39,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,42)(20,35,37,22)(21,30,36,33)(25,27,34,47)(26,40,44,43)(29,31,48,38)(39,45,46,41);(9,12,13,11)(10,14,15,16)(17,20,36,31,23,37,21,38)(18,25,44,45,28,34,26,41)(19,29,30,22,32,48,33,35)(24,39,40,47,42,46,43,27);(2,6)(5,7)(17,21,23,36)(18,26,28,44)(19,30,32,33)(20,31,37,38)(22,29,35,48)(24,40,42,43)(25,45,34,41)(27,46,47,39);(3,8)(5,7)(17,22,21,29,23,35,36,48)(18,27,26,46,28,47,44,39)(19,31,33,20,32,38,30,37)(24,41,43,34,42,45,40,25)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,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=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727262);