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 | |
2147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,6,8)(3,4,7,5)(10,13)(11,15)(14,16);(2,3)(4,5)(7,8)(10,14)(13,16);(9,10)(11,16)(12,14)(13,15);(1,3)(2,4)(5,8)(6,7);(1,4,6,5)(2,7,8,3)Polycyclic presentation:
2^9|a^2=h,e^2=h,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 | AGL(1,ℤ/8)2, C2×#6360185, #72339707, … |
| maximal normal subgroup of | AGL(1,ℤ/8)2, C2×#6360185, #72339707, … |
| maximal quotient of | #72279390, #72279595, #72279618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267381);