Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458519 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16319 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,7,10,22,3)(4,11,15,8,17,25,16,24)(5,18,19,27,20,12,21,9)(14,26,30,28,31,23,32,29);(1,3,6,10,7,13,22,2)(4,11,15,24,17,25,16,8)(5,12,19,27,20,18,21,9)(14,28,30,26,31,29,32,23);(1,4,6,15,7,17,22,16)(2,8,3,11,10,24,13,25)(5,14,19,30,20,31,21,32)(9,26,12,29,27,23,18,28);(1,5)(2,9)(3,12)(4,14)(6,19)(7,20)(8,23)(10,27)(11,28)(13,18)(15,30)(16,32)(17,31)(21,22)(24,26)(25,29)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365715);