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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,31)(21,25)(22,28)(23,27)(24,26)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,26)(21,23)(22,31)(24,30)(27,29)(28,32);(9,11)(10,13)(12,16)(14,15)(17,20,22,29,23,32,24,25)(18,21,26,30,27,19,28,31);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,19)(18,25,27,29)(20,28,32,26)(22,30,24,31);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,26,27,28)(19,30,21,31)(20,25,32,29)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 6936895);