Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,27)(19,28,30,41)(20,32,35,44)(21,25,34,26)(23,38,40,31)(24,36,43,37)(29,42,33,39)(45,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,30,34,38)(18,23,26,28,27,40,25,41)(20,33,37,46,35,29,36,47)(24,42,32,48,43,39,44,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,34,36,22,35,21,37)(18,24,25,44,27,43,26,32)(19,29,38,46,30,33,31,47)(23,39,41,48,40,42,28,45);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,30)(20,35)(23,40)(24,43)(28,41)(31,38)(32,44)(36,37);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,34)(18,25,27,26)(19,31,30,38)(20,36,35,37)(23,41,40,28)(24,32,43,44)(29,46,33,47)(39,45,42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461667);