Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201085 |
| 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 | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,8,4,3)(9,10,15,18,14,19,13,17)(11,20,22,16,23,24,21,12);(1,3,6,8,5,7,4,2)(9,11)(10,16)(12,19)(13,22)(14,23)(15,21)(17,24)(18,20);(1,4,5,6)(2,7,8,3)(9,12,14,16)(10,11,19,23)(13,20,15,24)(17,22,18,21);(1,4,5,6)(2,3,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108403);