Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,21)(23,24);(1,3)(2,6)(4,8)(5,7)(9,11,13,18)(10,14)(12,17,20,23)(15,21)(16,22)(19,24);(1,2)(3,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24);(1,4,6,5)(2,7,3,8)(10,16)(11,18)(15,19)(17,23)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105939);