Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 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)(17,18,23,28,24,20,22,27)(19,29,31,25,30,21,26,32);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19,24,30)(18,25,20,32)(21,28,29,27)(22,31,23,26);(1,4,5,3)(2,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,24)(19,25)(21,26)(22,27)(23,28)(29,31)(30,32);(9,11,13,16)(10,15,14,12)(17,21,24,29)(18,26,20,31)(19,27,30,28)(22,25,23,32);(9,12,13,15)(10,11,14,16)(19,30)(21,29)(25,32)(26,31)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=fg,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6530621);