Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3186 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(9,10)(11,14,18,22)(12,15)(13,16)(17,21,19,23)(20,24);(1,2,5,3,6,8,4,7)(9,11,13,18)(10,14)(12,19,20,17)(15,23)(16,22)(21,24);(1,3,5,8,6,7,4,2)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23);(12,20)(15,24)(17,19)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112974);