Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518936 |
| 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 | #59018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,30)(21,25)(23,27)(24,28)(26,32)(29,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,27)(18,22,28,23)(20,29,32,25)(21,30,31,26);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,31,24,32,22,21)(18,25,19,26,28,29,27,30);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,32,24,31,23,20)(18,26,27,25,28,30,19,29);(2,6)(7,8)(10,14)(13,15)(17,22,24,23)(18,19,28,27)(20,21,32,31)(25,26,29,30)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=fgi,d^2=fg,e^2=i,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=cgi,d^b=dh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7499643);