Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424733 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15324 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,8,6,11)(3,13,16,9,14,25,17,22)(4,18,19,10,20,26,21,23)(15,29,30,27,28,24,31,32);(2,8)(3,14)(5,6)(9,22)(15,28)(18,26)(19,21)(27,32);(1,3,5,16,7,14,6,17)(2,9,11,13,8,22,12,25)(4,15,19,30,20,28,21,31)(10,24,18,27,23,29,26,32);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,23)(9,24)(11,18)(12,26)(13,27)(14,28)(16,30)(17,31)(22,29)(25,32)Polycyclic presentation:
2^9|a^2=ef,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331929);