Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10631 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1445 |
| prime radical | C24 | C4⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,18,15,19,13,29)(11,20,23,33,24,32,22,16)(12,25,27,35,28,17,30,36)(21,37,38,31,39,34,26,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,23,15,24,13,22)(10,16,29,32,19,33,18,20)(12,26,27,21,28,38,30,39)(17,34,35,31,25,37,36,40);(1,2,4,6)(3,8,7,5)(9,12,13,30,15,28,14,27)(10,17,18,36,19,25,29,35)(11,21,22,26,24,39,23,38)(16,31,20,34,33,40,32,37)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fhi,d^c=dh,e^c=eh,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28906);