Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | #59654 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,24,29)(19,25,30,26)(20,31,21,32)(22,28,23,27);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,22)(21,23)(24,30)(26,29)(27,32)(28,31);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,23,30,24,21,22,19)(18,25,28,32,29,26,27,31);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,21,22,30,24,20,23,19)(18,26,27,32,29,25,28,31);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=fg,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=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505917);