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 | |
1641 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)(19,30)(20,25)(21,29)(22,38)(23,28)(24,33)(26,35)(27,32)(31,44)(34,47)(36,43)(37,42)(39,46)(40,45)(41,48);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,34)(21,32)(22,35)(23,33)(25,40)(26,27)(28,30)(29,38)(31,36)(37,45)(39,43)(41,44)(42,47)(46,48);(9,11,13,12)(10,14,15,16)(17,20,35,48,23,37,21,36)(18,25,26,41,28,42,29,43)(19,31,32,45,33,46,22,34)(24,39,38,47,30,44,27,40);(2,6)(5,7)(17,21,23,35)(18,26,28,29)(19,32,33,22)(20,36,37,48)(24,27,30,38)(25,41,42,43)(31,45,46,34)(39,40,44,47);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,22,21,33,23,32,35,19)(18,27,29,30,28,38,26,24)(20,34,36,46,37,45,48,31)(25,40,43,44,42,47,41,39)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=gh,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, 6741527);