Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | #25131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ 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)(4,8)(5,6)(17,18,23,28)(19,29,31,41)(20,34,38,43)(21,26,22,27)(24,33,40,32)(25,36,44,37)(30,39,35,42)(45,46,48,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,26,41,28,40,27,29)(20,35,37,46,38,30,36,47)(25,42,34,45,44,39,43,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,36,23,38,22,37)(18,25,27,43,28,44,26,34)(19,30,32,46,31,35,33,47)(24,39,29,45,40,42,41,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,38,37)(24,41,40,29)(25,34,44,43)(30,46,35,47)(39,48,42,45)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6463756);