Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301917 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×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 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23);(1,2,3,5,4,7,6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,22,23,24);(9,12)(10,15)(11,18,19,20)(13,21)(14,22,17,24)(16,23);(1,2,6,8,4,7,3,5)(10,16)(14,17)(15,23)(22,24)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=ehi,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 | #72268351, #72268358, #72268376, … |
| maximal normal subgroup of | #72268351, #72268358, #72268376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209137);