Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59041 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1761 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)(3,15)(4,11)(5,23)(6,10)(7,25)(8,14)(9,18)(12,22)(13,24)(16,26)(17,27)(19,29)(20,28)(21,31)(30,32);(1,3,7,18,8,20,24,19)(2,9,13,15,14,29,25,28)(4,21,6,16,22,32,5,17)(10,30,11,26,23,31,12,27);(1,4,7,6,8,22,24,5)(2,10,13,11,14,23,25,12)(3,16,19,21,20,17,18,32)(9,26,28,30,29,27,15,31);(1,5,7,4,8,6,24,22)(2,11,13,23,14,12,25,10)(3,17,18,21,20,16,19,32)(9,27,15,30,29,26,28,31);(1,6,7,22,8,5,24,4)(2,12,13,10,14,11,25,23)(3,16,18,32,20,17,19,21)(9,27,15,30,29,26,28,31)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,e^2=f,f^2=i,g^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,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442244);