Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16267 |
| 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 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,20,8)(3,12,15,23,13,22,16,7)(4,9,17,24,19,25,18,26)(14,29,31,32,28,27,30,21);(1,3,5,16,6,13,20,15)(2,7,10,22,11,23,8,12)(4,14,18,31,19,28,17,30)(9,21,26,29,25,32,24,27);(2,8)(3,13)(4,17)(5,20)(7,12)(9,25)(10,11)(14,30)(18,19)(22,23)(27,29)(28,31);(1,4)(2,9)(3,14)(5,18)(6,19)(7,21)(8,24)(10,26)(11,25)(12,27)(13,28)(15,30)(16,31)(17,20)(22,29)(23,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362840);