Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1291 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)(7,12,19,27)(8,20)(9,16)(10,15)(11,14)(13,22)(17,30,24,25)(18,32,34,35)(21,28)(23,29)(26,31,36,33);(1,2)(3,4)(7,17,19,24)(12,25,27,30)(18,33,34,31)(26,32,36,35);(2,4)(5,7)(6,12,14,27)(8,18)(9,24)(10,17)(11,19)(13,26,28,36)(15,30,16,25)(20,32,29,35)(21,31)(22,33)(23,34);(5,8,11,23)(6,13,14,28)(7,18,19,34)(9,22,10,21)(12,26,27,36)(15,20,16,29)(17,31,24,33)(25,35,30,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330622);