Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,11,8,10,14)(7,15,18,16,19,17,20,12);(1,2)(3,4)(5,7,10,18,11,19,9,20)(6,12,14,15,8,16,13,17);(2,4)(5,8,11,6)(7,16)(9,13,10,14)(12,19)(15,18)(17,20);(5,6,11,8)(7,17)(9,13,10,14)(12,18)(15,19)(16,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=efi,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284694, #72285492, #72285600, … |
| maximal normal subgroup of | #72284694, #72285492, #72285600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339207);