Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #39382 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1564 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)(19,28,30,24)(20,32,33,47)(21,26,23,37)(25,34,41,35)(29,43,45,46)(31,40,36,39)(38,44,42,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,40)(27,28)(32,46)(33,45)(34,48)(35,44)(37,39)(41,42)(43,47);(10,13)(14,16)(17,20,23,35,22,33,21,34)(18,25,26,32,27,41,37,47)(19,29,31,44,30,45,36,48)(24,38,40,46,28,42,39,43);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,30)(20,33)(24,28)(25,41)(31,36)(32,47)(34,35)(39,40);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,34,33,35)(24,39,28,40)(25,32,41,47)(29,44,45,48)(38,43,42,46)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466875);