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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,32,25)(20,26,35,44)(21,37,24,27)(22,28,38,40)(31,45,46,39)(33,43,47,41)(34,48,36,42);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,25,29,30)(20,33,35,47)(21,22,24,38)(26,41,44,43)(27,28,37,40)(31,34,46,36)(39,48,45,42);(2,6)(7,8)(10,14)(11,16)(17,20,21,34,23,35,24,36)(18,26,37,48,29,44,27,42)(19,31,38,47,32,46,22,33)(25,39,28,43,30,45,40,41);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,24)(18,27,29,37)(19,22,32,38)(20,34,35,36)(25,28,30,40)(26,42,44,48)(31,33,46,47)(39,43,45,41);(1,4,5,3)(2,8,6,7)(17,22,21,19,23,38,24,32)(18,28,37,30,29,40,27,25)(20,33,34,31,35,47,36,46)(26,43,48,45,44,41,42,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6748906);