Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| 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 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,10)(3,12,14,16)(4,11,6,8)(5,9,15,13)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,3,7,14)(2,5,10,15)(4,13,6,9)(8,12,11,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,4,7,6)(2,8,10,11)(3,13,14,9)(5,12,15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,5,7,15)(2,9,10,13)(3,11,14,8)(4,12,6,16);(2,10)(3,14)(8,11)(9,13)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339685, #72339717, #72339719, … |
| maximal normal subgroup of | #72339685, #72339717, #72339719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271493);