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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1426 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,39)(31,41)(33,40)(35,42)(36,44)(38,43)(45,46)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,33,21,31)(18,24,26,29,28,41,27,40)(20,35,38,46,36,30,37,47)(25,42,44,48,43,39,34,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,37,23,36,22,38)(18,25,27,34,28,43,26,44)(19,30,31,46,33,35,32,47)(24,39,40,48,41,42,29,45);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,36)(25,43)(30,35)(34,44)(37,38)(39,42)(45,48)(46,47);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,37,36,38)(24,29,41,40)(25,44,43,34)(30,46,35,47)(39,45,42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461538);