Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483993 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | #33605 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
603 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)(10,16)(11,20,24,22)(12,25)(13,26)(14,27)(17,31,33,34)(18,28)(19,30)(21,32,37,40)(23,38,35,36);(1,3)(2,5)(4,8)(6,7)(9,10,13,16,15,19,26,30)(11,21,35,17,24,37,23,33)(12,18,25,29,14,28,27,39)(20,34,38,40,22,31,36,32);(3,8)(5,7)(9,11,14,20,13,23,27,36,15,24,12,22,26,35,25,38)(10,17,18,32,16,21,29,31,19,33,28,40,30,37,39,34);(2,6)(5,7)(9,12,13,25,15,14,26,27)(10,18,16,29,19,28,30,39)(11,22,23,38,24,20,35,36)(17,32,21,31,33,40,37,34)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=h,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264749, #72264753, #72264790, … |
| maximal normal subgroup of | #72264749, #72264753, #72264790, … |
| maximal quotient of | #72320578, #72320588, C2×#497030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404226);