Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #25123 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1435 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)(3,7)(4,8)(5,6)(17,18,24,21)(19,29,31,20)(22,27,23,28)(25,30,32,26);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30,24,31,22,26)(18,25,27,20,21,32,28,29);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,32,24,29,22,25)(18,26,27,19,21,30,28,31);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(17,21,24,18)(19,25,31,32)(20,26,29,30)(22,27,23,28);(11,15)(14,16)(17,22,24,23)(18,27,21,28)(19,30,31,26)(20,25,29,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6463730);