Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1519 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,26)(19,24,29,23)(20,25,22,27)(28,32,30,31);(2,5)(6,8)(19,28)(20,22)(23,31)(24,32)(25,27)(29,30);(9,11,12,16)(10,13,14,15)(17,19,21,29)(18,23,26,24)(20,28,22,30)(25,31,27,32);(17,19,21,29)(18,24,26,23)(20,28,22,30)(25,32,27,31);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,25,26,27)(19,28,29,30)(23,31,24,32)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6353575);