Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16308 |
| 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 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,17)(5,7)(9,22)(10,25)(14,29)(15,16)(18,19)(20,24)(21,27)(31,32);(1,2,5,11,6,8,7,12)(3,9,13,24,16,20,15,22)(4,10,17,27,18,21,19,25)(14,23,28,32,29,26,30,31);(1,3,5,13,6,16,7,15)(2,9,12,22,8,20,11,24)(4,14,17,28,18,29,19,30)(10,26,25,32,21,23,27,31);(1,4)(2,10)(3,14)(5,17)(6,18)(7,19)(8,21)(9,23)(11,27)(12,25)(13,28)(15,30)(16,29)(20,26)(22,31)(24,32)Polycyclic presentation:
2^9|b^2=f,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,g^c=gi,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, 365222);