Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,16,10,17,11,15)(7,13,22,33,20,34,21,32)(8,23,25,28,26,29,14,9)(18,27,24,36,31,19,35,30);(5,7,10,20)(6,13,17,34)(8,24,26,35)(9,27,28,19)(11,22,12,21)(14,31,25,18)(15,33,16,32)(23,30,29,36);(2,4)(5,8,11,25,10,26,12,14)(6,9,16,29,17,28,15,23)(7,18,21,24,20,31,22,35)(13,30,33,19,34,36,32,27);(2,4)(5,9)(6,14)(7,19)(8,16)(10,28)(11,29)(12,23)(13,31)(15,26)(17,25)(18,34)(20,27)(21,36)(22,30)(24,32)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295570);