Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361928 |
| 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 | C2×#135 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1516 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18,23,28)(19,29,31,32)(20,30,24,25)(21,27,22,26);(3,10)(6,15)(9,16)(11,13)(17,19,23,31)(18,24,28,20)(21,25,22,30)(26,32,27,29);(1,3,5,10)(2,6,7,15)(4,13,14,11)(8,16,12,9)(17,20)(18,19)(21,32)(22,29)(23,24)(25,27)(26,30)(28,31);(2,7)(6,15)(8,12)(9,16)(17,20)(18,25)(19,27)(21,29)(22,32)(23,24)(26,31)(28,30);(17,21,23,22)(18,26,28,27)(19,25,31,30)(20,29,24,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269171);