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 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,16,10,15)(7,17,19,29,21,28,20,12)(8,22,25,32,24,31,26,13)(18,33,35,36,34,27,23,30);(1,2)(3,4)(5,7,9,20,11,21,10,19)(6,12,14,28,16,29,15,17)(8,23,25,34,24,35,26,18)(13,30,22,27,32,36,31,33);(2,4)(5,8,10,26,11,24,9,25)(6,13,14,22,16,32,15,31)(7,18,20,23,21,34,19,35)(12,27,17,30,29,33,28,36);(6,14,16,15)(8,24)(12,28,29,17)(13,31,32,22)(18,34)(23,35)(25,26)(27,30,33,36)Polycyclic presentation:
2^9|a^2=e,b^2=efh,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,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, 316369);