Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2006 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,15,12,13,17,14,22)(11,18,20,16,21,19,23,24);(1,3,4,8)(2,5,6,7)(9,10,14,12,13,17,15,22)(11,18,20,24,21,19,23,16);(2,6)(5,7)(9,11)(10,16)(12,19)(13,21)(14,23)(15,20)(17,24)(18,22);(3,8)(5,7)(9,12,15,17,13,22,14,10)(11,19,20,24,21,18,23,16)Polycyclic presentation:
2^9|a^2=eg,b^2=f,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343906);