Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,9,11,15)(4,13,14,8)(6,10,16,12)(17,18,21,25)(19,26,28,31)(20,24,22,29)(23,30,32,27);(1,3)(2,6)(4,10)(5,11)(7,16)(8,15)(9,13)(12,14)(17,18)(19,23)(20,29)(21,25)(22,24)(26,27)(28,32)(30,31);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,19,21,28)(18,23,25,32)(20,27,22,30)(24,31,29,26);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,32,26);(2,7)(3,11)(8,13)(10,12)(18,25)(23,32)(24,29)(26,31)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339693, #72339707, #72339736, … |
| maximal normal subgroup of | #72339693, #72339707, #72339736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275659);