Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,17,11,15)(7,18,21,28,14,29,20,9)(8,22,25,33,27,32,26,13)(19,24,35,30,31,36,23,34);(1,2)(3,4)(5,7,10,21,12,14,11,20)(6,9,16,18,17,28,15,29)(8,23,25,19,27,35,26,31)(13,30,22,36,33,34,32,24);(2,4)(5,8,12,27)(6,13,17,33)(7,19,14,31)(9,24,28,36)(10,26,11,25)(15,22,16,32)(18,34,29,30)(20,35,21,23);(5,9,12,28)(6,14)(7,17)(8,24,27,36)(10,18,11,29)(13,31)(15,21)(16,20)(19,33)(22,23)(25,30,26,34)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 337539);