Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(17,18)(19,20)(21,25)(22,26)(23,32)(24,27)(28,30)(29,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,24,29)(18,20,27,31)(21,23,28,22)(25,32,30,26);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,23,25,24,31,22,30)(18,21,26,19,27,28,32,29);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21,24,28)(18,25,27,30)(19,22,29,23)(20,26,31,32);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,22,24,23)(18,26,27,32)(19,28,29,21)(20,30,31,25)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6467775);