Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421159 |
| 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 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,12,17,11,16)(7,18,21,33,23,32,22,13)(8,24,26,35,9,28,27,14)(19,31,25,34,20,30,29,36);(1,2)(3,4)(5,7,10,22,12,23,11,21)(6,13,15,32,17,33,16,18)(8,25,26,19,9,29,27,20)(14,34,24,31,35,36,28,30);(2,4)(5,8,12,9)(6,14,17,35)(7,19)(10,27,11,26)(13,30)(15,28,16,24)(18,34)(20,23)(21,29)(22,25)(31,33)(32,36);(5,9,12,8)(6,14,17,35)(7,20)(10,27,11,26)(13,31)(15,24,16,28)(18,34)(19,23)(21,29)(22,25)(30,33)(32,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,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, 339196);