Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| 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 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,18)(8,21)(9,23,26,31)(10,16)(11,17)(12,13)(14,24,32,25)(15,27)(19,28,35,36)(20,29)(22,34,33,30);(1,2,3,4)(5,7,10,13,11,21,27,20)(6,12,15,18,17,29,16,8)(9,19,25,30,26,35,24,34)(14,28,23,33,32,36,31,22);(5,8,10,12,11,18,27,29)(6,13,15,21,17,20,16,7)(9,22,25,28,26,33,24,36)(14,30,23,35,32,34,31,19);(5,9)(6,14)(7,19)(8,22)(10,25)(11,26)(12,28)(13,30)(15,23)(16,31)(17,32)(18,33)(20,34)(21,35)(24,27)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300286);