Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,11,15,22,16)(7,13,19,29,20,12,8,21)(9,24,26,14,27,32,25,31)(18,33,23,30,35,36,34,28);(1,2)(3,4)(5,7,11,20)(6,12,15,13)(8,22,19,10)(9,18,27,35)(14,30,31,28)(16,21,17,29)(23,26,34,25)(24,36,32,33);(2,4)(5,8,10,7,11,19,22,20)(6,13,16,21,15,12,17,29)(9,23,25,18,27,34,26,35)(14,30,32,36,31,28,24,33);(5,9)(6,14,15,31)(7,18)(8,23)(10,25)(11,27)(12,28,13,30)(16,32,17,24)(19,34)(20,35)(21,36,29,33)(22,26)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291537);