Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10,12,15)(11,13,16,14)(17,18,21,23,20,19,22,24);(1,2,5,6)(3,7,4,8)(17,19,21,24,20,18,22,23);(1,3,5,4)(2,7,6,8)(10,13)(11,16)(14,15)(18,19)(23,24);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 137919);