Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363930 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,3)(4,12,14,16)(5,9,6,11)(8,15,10,13)(18,24)(19,26)(21,22)(25,32)(27,28)(29,30);(1,3,7,2)(4,10,14,8)(5,9,6,11)(12,13,16,15)(17,18)(19,23)(20,28)(21,26)(22,24)(25,30)(27,29)(31,32);(1,4)(2,8)(3,10)(5,13)(6,15)(7,14)(9,16)(11,12)(17,19,23,18)(20,27,31,25)(21,26,22,24)(28,30,32,29);(17,20,23,31)(18,25,19,27)(21,29,22,30)(24,32,26,28);(1,5,7,6)(2,9,3,11)(4,13,14,15)(8,16,10,12)(17,21,23,22)(18,24,19,26)(20,29,31,30)(25,32,27,28)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271429);