Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | #36213 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,24,30,28)(21,25,22,26)(27,31,29,32);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,23,30)(18,24,20,28)(21,29,22,27)(25,32,26,31);(10,14)(13,16)(19,27)(21,22)(24,31)(25,26)(28,32)(29,30);(2,6)(5,7)(17,20,23,18)(19,28,30,24)(21,26,22,25)(27,32,29,31);(3,8)(5,7)(17,21)(18,25)(20,26)(22,23)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6585114);