Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,19)(11,21,18,17)(12,15,20,16)(13,24,22,23);(1,3)(2,6)(4,7)(5,8)(9,11,15,24,14,18,16,23)(10,17,20,13,19,21,12,22);(1,2,5,4)(3,6,8,7)(9,12,15,19,14,20,16,10)(11,22,23,17,18,13,24,21);(1,4,5,2)(3,6,8,7)(9,13,14,22)(10,18,19,11)(12,24,20,23)(15,21,16,17)Polycyclic presentation:
2^9|a^2=e,b^2=fg,c^2=efi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346460);