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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,15)(12,13)(16,23)(17,18)(19,24)(20,22);(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,23)(22,24);(9,11)(10,16)(12,20)(13,22)(14,21)(15,19)(17,24)(18,23);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224743, #72224757, #72224758, … |
| maximal normal subgroup of | #72224743, #72224757, #72224758, … |
| maximal quotient of | #72257886, #72264295 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321879);