Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #59041 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,9,4)(3,13,18,14)(5,21,24,28)(6,7,12,8)(10,17,19,16)(11,25,20,23)(15,26,31,32)(22,29,30,27);(1,3,7,16,9,18,8,17)(2,10,6,13,4,19,12,14)(5,22,23,15,24,30,25,31)(11,27,21,26,20,29,28,32);(1,4,7,12,9,2,8,6)(3,14,17,19,18,13,16,10)(5,20,23,28,24,11,25,21)(15,27,22,32,31,29,30,26);(1,5,7,23,9,24,8,25)(2,11,6,21,4,20,12,28)(3,15,16,30,18,31,17,22)(10,26,13,29,19,32,14,27);(1,6,7,4,9,12,8,2)(3,13,16,19,18,14,17,10)(5,21,23,20,24,28,25,11)(15,29,30,32,31,27,22,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442252);