Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3543 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,10,11,15)(4,8,13,12)(5,14,16,9)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,6,11)(2,4,7,13)(5,8,16,12)(9,15,14,10)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32);(2,7)(3,11)(8,12)(9,14)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,6,13)(2,8,7,12)(3,9,11,14)(5,15,16,10);(2,7)(3,11)(8,12)(9,14)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283751, #72306907, #72306917, … |
| maximal normal subgroup of | #72283751, #72306907, #72306917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274463);