Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6489185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35237 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,11,7,12,6,8)(3,9,15,13,17,25,16,22)(4,18,19,10,21,23,20,26)(14,27,28,24,30,31,29,32);(1,2)(3,13)(4,10)(5,8)(6,11)(7,12)(9,15)(14,27)(16,25)(17,22)(18,19)(20,23)(21,26)(24,29)(28,32)(30,31);(2,8,12,11)(9,22,25,13)(10,23,26,18)(24,31,32,27);(1,3,6,16,7,17,5,15)(2,9,8,22,12,25,11,13)(4,14,19,28,21,30,20,29)(10,24,23,31,26,32,18,27);(1,4,6,19,7,21,5,20)(2,10,11,18,12,26,8,23)(3,14,16,28,17,30,15,29)(9,24,13,27,25,32,22,31)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396383);