Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | |
1465 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)(17,18,21,24)(19,26,25,22)(20,27,28,32)(23,30,31,29);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31);(3,8)(5,7)(10,13)(14,15)(17,20,21,28)(18,23,24,31)(19,27,25,32)(22,30,26,29);(2,6)(5,7)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32);(9,11,12,16)(10,14,13,15)(18,24)(19,25)(23,31)(27,32)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707537);