Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476441 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17089 |
| 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 | |
836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,12)(3,13,17,24,14,28,16,8)(4,18,22,9,21,26,20,27)(15,23,19,25,30,32,31,29);(1,3,5,16,6,14,7,17)(2,8,10,13,11,24,12,28)(4,19,20,15,21,31,22,30)(9,25,26,32,27,29,18,23);(3,14)(4,20)(5,7)(9,26)(10,12)(13,28)(15,19)(18,27)(21,22)(23,25)(29,32)(30,31);(1,4)(2,9)(3,15)(5,22)(6,21)(7,20)(8,23)(10,18)(11,27)(12,26)(13,29)(14,30)(16,19)(17,31)(24,32)(25,28)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383637);