Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(4,8)(5,7)(9,10,12,13)(11,14,16,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,4,5,3)(2,6,7,8)(10,13)(11,15)(14,16);(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31);(10,13)(11,15)(14,16)(19,27)(22,25)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6598101);