Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,12,18,14,17,13,16)(11,19,22,24,21,15,20,23);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,14,21,12,20)(10,15,16,23,17,19,18,24);(3,7)(5,8)(11,20,21,22)(15,23,19,24);(10,16,17,18)(15,24,19,23)Polycyclic presentation:
2^9|a^2=eg,b^2=efh,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
| maximal quotient of | #72235145, #72235150, #72235153 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361000);