Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 | #59410 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,30,20,25)(21,32,31,26)(22,29,24,27);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,25,28,30)(21,24,31,22)(26,27,32,29);(10,13)(14,15)(17,20,24,31,23,19,22,21)(18,26,29,25,28,32,27,30);(9,11,12,16)(10,14,13,15)(17,21,24,20,23,31,22,19)(18,25,29,32,28,30,27,26);(1,3,4,8)(2,6,5,7)(17,22,23,24)(18,27,28,29)(19,21,20,31)(25,32,30,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503859);