Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| 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 | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1005 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)(7,12,18,28)(8,21,24,29)(9,16)(10,14)(11,15)(13,23,30,25)(17,32)(19,27,20,31)(22,35)(26,34)(33,36);(5,7,9,20,11,18,10,19)(6,12,14,27,15,28,16,31)(8,22,23,33,24,17,25,34)(13,26,29,32,30,36,21,35);(5,8)(6,13,15,30)(7,17,18,22)(9,23)(10,25)(11,24)(12,26)(14,29,16,21)(19,33,20,34)(27,32)(28,36)(31,35);(2,4)(6,14)(7,18)(9,10)(12,27)(13,29)(15,16)(17,22)(21,30)(23,25)(26,32)(28,31)(35,36)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=e,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,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340211);