Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,29,12,17,11,16)(7,18,30,32,22,31,21,13)(8,9,25,27,15,14,24,28)(19,26,34,20,35,33,23,36);(5,7,10,30,12,22,11,21)(6,13,29,18,17,32,16,31)(8,23,24,35,15,34,25,19)(9,26,28,36,14,33,27,20);(5,8)(6,14)(7,19)(9,17)(10,24)(11,25)(12,15)(13,20)(16,28)(18,26)(21,34)(22,35)(23,30)(27,29)(31,33)(32,36);(2,4)(5,9)(6,15)(7,20)(8,17)(10,28)(11,27)(12,14)(13,19)(16,24)(18,23)(21,33)(22,36)(25,29)(26,30)(31,34)(32,35)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340658);