Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42529 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,13,15)(11,14,12,16);(1,2)(3,6)(4,8)(5,7)(9,11,13,12)(10,14,15,16)(17,18,22,24)(19,26,27,29)(20,28,23,31)(21,30,32,25);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,24)(21,31)(22,27)(25,29)(26,30)(28,32);(1,3,5,4)(2,6,7,8)(18,24)(19,27)(25,30)(28,31);(1,4,5,3)(2,6,7,8)(17,20,22,23)(18,19,24,27)(21,26,32,29)(25,31,30,28)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6601641);