Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394038 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1680 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)(10,15)(11,18)(12,13)(16,23)(17,22)(20,21);(1,3,4,7)(2,5,6,8)(9,10)(11,19)(12,22)(13,15)(14,17)(16,20)(18,23)(21,24);(9,11,12,20,14,21,13,18)(10,16,15,19,17,23,22,24);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224743, #72224750, #72252526, … |
| maximal normal subgroup of | #72224743, #72224750, #72252526, … |
| maximal quotient of | #72257811, #72264379 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317181);