Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
769 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)(5,6,9,16,11,15,10,14)(7,17,19,29,20,12,26,28)(8,21,24,13,23,31,25,32)(18,34,22,33,35,30,36,27);(5,7,10,19,11,20,9,26)(6,12,14,28,15,17,16,29)(8,22,25,35,23,36,24,18)(13,30,21,27,32,34,31,33);(2,4)(5,8,9,24,11,23,10,25)(6,13,14,21,15,32,16,31)(7,18,19,22,20,35,26,36)(12,27,29,30,17,33,28,34);(6,14,15,16)(8,23)(12,28,17,29)(13,31,32,21)(18,35)(22,36)(24,25)(27,30,33,34)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=g,d^2=e,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=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317606);