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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1409 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)(11,14)(12,13)(15,16)(17,18)(19,30)(20,33)(21,38)(22,39)(23,29)(24,27)(25,32)(26,36)(28,40)(31,41)(34,42)(35,43)(37,47)(44,45)(46,48);(1,3,4,8)(2,5,6,7)(17,19)(18,25)(20,34)(21,22)(23,32)(24,40)(26,42)(27,39)(28,38)(29,30)(31,37)(33,46)(35,45)(36,48)(41,43)(44,47);(10,13)(14,16)(17,20,24,37,23,36,21,35)(18,26,27,43,29,33,38,47)(19,31,22,34,32,45,40,48)(25,41,28,42,30,44,39,46);(2,6)(5,7)(17,21,23,24)(18,27,29,38)(19,22,32,40)(20,35,36,37)(25,39,30,28)(26,43,33,47)(31,34,45,48)(41,46,44,42);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,24,32,23,40,21,19)(18,28,27,30,29,39,38,25)(20,34,37,45,36,48,35,31)(26,42,43,44,33,46,47,41)Polycyclic presentation:
2^9|b^2=fi,c^2=h,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6737261);