Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14823 |
| 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 | |
1189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,21,8)(3,12,15,23,16,7,13,24)(4,17,19,9,20,26,18,25)(14,29,30,22,31,32,28,27);(1,3)(2,7)(4,14)(5,13)(6,16)(8,24)(9,22)(10,12)(11,23)(15,21)(17,32)(18,30)(19,28)(20,31)(25,27)(26,29);(2,8)(3,13)(5,21)(9,17)(10,11)(14,28)(15,16)(18,19)(23,24)(25,26)(29,32)(30,31);(1,4)(2,9,10,25)(3,14)(5,19)(6,20)(7,22,12,27)(8,17,11,26)(13,28)(15,30)(16,31)(18,21)(23,29,24,32)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216130, #72216217, … |
| maximal normal subgroup of | #72216047, #72216130, #72216217, … |
| maximal quotient of | C2×#401792, #72252170, #72255599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308988);