Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,3,5,7,8)(10,15)(11,20)(13,21)(17,22)(18,19)(23,24);(2,5)(6,8)(9,10,14,19)(11,16,22,12)(13,18,21,15)(17,24,20,23);(9,11,21,17,14,22,13,20)(10,16,15,24,19,12,18,23);(2,5)(6,8)(9,12)(10,17)(11,18)(13,24)(14,16)(15,22)(19,20)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dei,e^b=eh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132698);