Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1057 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)(5,6,10,12,11,16,9,15)(7,17,20,30,18,27,26,13)(8,21,24,32,25,28,23,14)(19,29,22,31,34,36,35,33);(6,12)(7,18)(9,10)(13,27)(14,28)(15,16)(17,30)(21,32)(22,35)(23,24)(29,33)(31,36);(5,7,9,20,11,18,10,26)(6,13,15,17,16,30,12,27)(8,22,23,34,25,35,24,19)(14,31,28,36,32,33,21,29);(2,4)(5,8,11,25)(6,14,16,32)(7,19,18,34)(9,24,10,23)(12,28,15,21)(13,29,30,36)(17,33,27,31)(20,35,26,22)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383677);