Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394038 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2412 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)(11,18)(12,13)(15,19)(16,22)(20,21)(23,24);(1,3,4,7)(2,5,6,8)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,23)(20,24);(9,11)(10,15)(12,21)(13,18)(14,20)(16,24)(17,23)(19,22);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224743, #72224752, #72252474, … |
| maximal normal subgroup of | #72224743, #72224752, #72252474, … |
| maximal quotient of | #72257799, #72260723, #72262056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312298);