Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1500 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(18,23)(24,26)(25,30)(31,32);(1,3,5,4)(2,6,8,7)(18,24)(19,27)(20,22)(23,26)(28,29)(31,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,18,21,23)(19,25,28,30)(20,24,22,26)(27,31,29,32);(1,4,5,3)(2,6,8,7)(17,19)(18,25)(20,27)(21,28)(22,29)(23,30)(24,31)(26,32);(17,20,21,22)(18,26,23,24)(19,27,28,29)(25,32,30,31)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353453);