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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,28)(19,29,31,41)(20,34,36,44)(21,27,22,26)(24,33,40,32)(25,37,43,38)(30,39,35,42)(45,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,27,41,28,40,26,29)(20,35,38,46,36,30,37,47)(25,42,34,45,43,39,44,48);(10,13)(14,16)(17,20,21,37,23,36,22,38)(18,25,26,44,28,43,27,34)(19,30,32,46,31,35,33,47)(24,39,29,45,40,42,41,48);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,31)(20,36)(24,40)(25,43)(29,41)(32,33)(34,44)(37,38);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=fg,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,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466171);