Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45319 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1638 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,19,23,28,22,27)(20,21,31,30,32,25,26,29);(9,10)(11,15)(12,14)(13,16)(17,19,23,27)(18,22,28,24)(20,30,32,29)(21,26,25,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,32)(18,25,28,21)(19,29,27,30)(22,26,24,31);(11,16)(13,15)(17,21,24,30,23,25,22,29)(18,26,19,20,28,31,27,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,27,28,19)(20,31,32,26)(21,30,25,29)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=ghi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6998361);