Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449963 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15954 |
| 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 | |
1042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,20)(3,12,15,24,14,8,21,25)(4,16,18,27,19,9,22,28)(13,29,31,32,30,23,17,26);(1,3)(2,8,10,12)(4,17)(5,21)(6,15)(7,14)(9,26,16,32)(11,25,20,24)(13,18)(19,31)(22,30)(23,28,29,27);(1,4)(2,9)(3,13,14,30)(5,18)(6,22)(7,19)(8,23,12,29)(10,16)(11,27)(15,17,21,31)(20,28)(24,26,25,32);(1,2)(3,12)(4,9)(5,11)(6,20)(7,10)(8,14)(13,29)(15,25)(16,19)(17,32)(18,27)(21,24)(22,28)(23,30)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237078, #72237090, … |
| maximal normal subgroup of | #72237064, #72237078, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357159);