Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| 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×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 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)(7,17,18,29,20,27,26,13)(8,21,24,32,25,28,23,14)(19,30,22,31,35,36,34,33);(6,12)(7,18)(9,10)(13,29)(14,28)(15,16)(19,22)(20,26)(21,32)(23,24)(31,33)(34,35);(5,7,9,18,11,20,10,26)(6,13,15,17,16,29,12,27)(8,22,23,35,25,34,24,19)(14,31,28,36,32,33,21,30);(2,4)(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,28)(13,30)(15,21)(16,32)(17,33)(18,34)(20,35)(22,26)(27,31)(29,36)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,g^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,g^b=gi,h^b=hi,d^c=de,f^c=fi,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, 383682);