Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1511 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)(9,10,12,16,14,11,13,15)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31);(9,10,14,11)(12,15,13,16)(19,26)(22,25)(27,32)(29,31);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,15,11,16)(20,28)(23,30)(27,32)(29,31);(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=i,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6463241);