Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360760 |
| 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 | #61288 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1536 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)(3,10)(4,8)(5,13)(6,9)(7,12)(11,16)(14,15)(17,18,24,29)(19,30,25,32)(20,31,21,26)(22,28,23,27);(3,11)(7,15)(10,16)(12,14)(17,19,24,25)(18,20,29,21)(22,31,23,26)(27,32,28,30);(1,3)(2,7)(4,12)(5,14)(6,11)(8,16)(9,15)(10,13)(17,20,24,21)(18,25,29,19)(22,30,23,32)(26,27,31,28);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(17,21,24,20)(18,26,29,31)(19,28,25,27)(22,30,23,32);(17,22,24,23)(18,27,29,28)(19,31,25,26)(20,32,21,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268015);