Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,17,10,27)(7,18,22,30,8,14,20,13)(9,23,28,34,26,15,25,33)(19,35,24,29,21,31,36,32);(1,2)(3,4)(5,7,10,20,11,8,12,22)(6,13,16,18,17,30,27,14)(9,24,25,19,26,36,28,21)(15,31,33,32,23,35,34,29);(2,4)(5,8)(6,14)(7,11)(9,24)(10,20)(12,22)(13,27)(15,32)(16,30)(17,18)(19,28)(21,25)(23,29)(26,36)(31,33)(34,35);(5,9,11,26)(6,15,17,23)(7,19,8,21)(10,28,12,25)(13,29,30,32)(14,31,18,35)(16,34,27,33)(20,24,22,36)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=i,e^2=i,g^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,d^b=def,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383636);