Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301917 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,15)(11,17,20,24)(12,21,18,16)(14,22,23,19);(1,3)(2,5)(4,8)(6,7)(9,11,12,19,13,20,18,22)(10,14,16,17,15,23,21,24);(1,2)(3,7,8,5)(4,6)(10,15)(14,23)(16,21)(17,24);(9,11,18,22,13,20,12,19)(10,14,21,24,15,23,16,17)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268345, #72268351, #72268369, … |
| maximal normal subgroup of | #72268345, #72268351, #72268369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209120);