Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459444 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,18,10,17)(7,19,22,15,9,27,21,13)(8,14,26,31,25,33,24,32)(20,35,36,29,28,30,23,34);(1,2)(3,4)(5,7,11,9)(6,13)(8,23,25,36)(10,21,12,22)(14,30)(15,18)(16,19)(17,27)(20,24,28,26)(29,32)(31,34)(33,35);(2,4)(5,8,11,25)(6,14,18,33)(7,20,9,28)(10,26,12,24)(13,29,15,34)(16,32,17,31)(19,35,27,30)(21,36,22,23);(5,9,11,7)(6,15)(8,23,25,36)(10,21,12,22)(13,18)(14,30)(16,19)(17,27)(20,26,28,24)(29,31)(32,34)(33,35)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,d^2=f,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,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366642);