Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1498 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)(11,14)(12,16)(13,15)(17,18,24,29,23,28,22,27)(19,30,21,25,33,40,32,41)(20,34,38,43,37,26,36,44)(31,42,35,48,47,45,46,39);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,35)(21,22)(23,33)(24,32)(26,42)(27,40)(28,41)(29,30)(31,38)(34,45)(36,47)(37,46)(39,44)(43,48);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,24,38,23,37,22,36)(18,26,29,44,28,34,27,43)(19,31,21,35,33,47,32,46)(25,39,40,42,41,48,30,45);(1,4,5,3)(2,8,6,7)(17,21,22,19,23,32,24,33)(18,25,27,30,28,41,29,40)(20,35,36,31,37,46,38,47)(26,39,43,45,34,48,44,42);(17,22,23,24)(18,27,28,29)(19,21,33,32)(20,36,37,38)(25,40,41,30)(26,43,34,44)(31,35,47,46)(39,42,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365671);