Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483919 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,21,32,20)(11,16,13,28,23,18,30,34)(12,24,26,39,27,35,31,33)(17,29,36,22,38,25,37,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,13,15,23,32,30)(10,16,19,28,21,18,20,34)(12,25,26,40,27,29,31,22)(17,35,36,33,38,24,37,39);(3,8)(5,7)(9,12,15,27)(10,17,21,38)(11,22,23,40)(13,29,30,25)(14,31,32,26)(16,33,18,39)(19,37,20,36)(24,28,35,34);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265270, #72265276, #72265283, … |
| maximal normal subgroup of | #72265270, #72265276, #72265283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404033);