Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
1293 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)(5,6,11,18,12,16,10,17)(7,19,23,31,24,13,22,32)(8,25,28,9,27,29,15,14)(20,35,30,21,26,36,34,33);(5,7,12,24)(6,13,16,19)(8,26,27,20)(9,21,14,33)(10,23,11,22)(15,34,28,30)(17,32,18,31)(25,36,29,35);(2,4)(5,8,11,15,12,27,10,28)(6,14,17,25,16,9,18,29)(7,20,23,34,24,26,22,30)(13,21,31,36,19,33,32,35);(2,4)(5,9)(6,15)(7,21)(8,18)(10,25)(11,29)(12,14)(13,30)(16,28)(17,27)(19,34)(20,31)(22,35)(23,36)(24,33)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,f^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,h^b=hi,e^c=ei,g^c=gi,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, 295564);