Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,15,10,13)(7,12,19,25,17,28,23,27)(8,20,21,14,22,29,24,30)(18,32,33,35,31,36,34,26);(1,2)(3,4)(5,7,9,23,11,17,10,19)(6,12,13,25,15,28,16,27)(8,18,21,34,22,31,24,33)(14,26,20,32,30,35,29,36);(2,4)(6,13)(7,17)(9,10)(12,25)(14,29)(15,16)(20,30)(21,24)(26,32)(27,28)(33,34)(35,36);(5,8)(6,14)(7,18)(9,21)(10,24)(11,22)(12,26)(13,20)(15,30)(16,29)(17,31)(19,33)(23,34)(25,32)(27,36)(28,35)Polycyclic presentation:
2^9|a^2=f,b^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363866);