Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #25771 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,17,19,14)(12,20,21,15)(13,16)(18,22)(23,24);(1,2,4,7)(3,5,8,6)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(2,5)(3,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72268293, #72268295, #72268308, … |
| maximal normal subgroup of | #72268293, #72268295, #72268308, … |
| maximal quotient of | #72313370, #72313400 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174577);