Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360176 |
| 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 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10)(3,8,12,15)(4,9,5,7)(11,16,13,14)(17,18,22,25)(19,26,27,32)(20,24,21,29)(23,30,31,28);(2,7)(3,11)(4,5)(8,15)(9,10)(12,13)(19,27)(23,31)(26,32)(28,30);(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,31)(26,29);(1,3,6,12)(2,8,10,15)(4,11,5,13)(7,14,9,16);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,16,15,14)(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=hi,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306653, #72306661, #72306662, … |
| maximal normal subgroup of | #72306653, #72306661, #72306662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267547);