Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,18,10,16,11,17)(7,8,14,24,21,13,20,23)(9,15,27,32,25,30,26,31)(19,28,33,35,34,22,29,36);(1,2)(3,4)(5,7)(6,13)(8,16)(9,19)(10,21)(11,14)(12,20)(15,22)(17,23)(18,24)(25,34)(26,33)(27,29)(28,30)(31,36)(32,35);(2,4)(5,8,10,13)(6,14)(7,17)(9,22,25,28)(11,23,12,24)(15,29)(16,20)(18,21)(19,32)(26,35,27,36)(30,33)(31,34);(5,9,10,25)(6,15,16,30)(7,19,21,34)(8,22,13,28)(11,27,12,26)(14,29,20,33)(17,32,18,31)(23,36,24,35)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^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,f^c=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291461);