Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,14,10,13,9,12)(7,15,17,20,18,11,16,19);(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,15)(14,20);(2,4)(6,12)(7,16)(8,9)(13,14)(17,18)(19,20);(6,12,13,14)(11,19,15,20)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338466, #72338487, #72338490, … |
| maximal normal subgroup of | #72338466, #72338487, #72338490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367503);