Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35840 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1072 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)(14,18)(19,22)(20,24,21,23);(1,2,4,5)(3,7,8,6)(9,11,14,12)(10,15,18,17)(13,21,22,20)(16,24,19,23);(1,3,4,8)(2,6,5,7)(10,16)(13,22)(15,23)(17,24)(18,19)(20,21);(1,2,4,5)(3,6,8,7)(9,12,14,11)(10,17,18,15)(13,21,22,20)(16,24,19,23)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fi,g^d=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121930);