Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,14,9,16,10,15)(7,12,24,28,18,29,19,27)(8,20,25,31,22,32,23,13)(17,30,34,36,33,35,21,26);(5,7)(6,12)(8,21)(9,18)(10,24)(11,19)(13,30)(14,27)(15,28)(16,29)(17,23)(20,26)(22,34)(25,33)(31,35)(32,36);(1,2,3,4)(5,8,10,23,9,22,11,25)(6,13,14,20,16,31,15,32)(7,17,19,21,18,33,24,34)(12,26,28,30,29,36,27,35);(6,14,16,15)(8,22)(12,27,29,28)(13,20,31,32)(17,33)(21,34)(23,25)(26,35,36,30)Polycyclic presentation:
2^9|a^2=eg,c^2=f,d^2=ei,f^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,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368817);