Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(9,10)(11,15,16,13)(12,14);(2,6)(4,7)(9,10)(11,13)(12,14)(15,16);(1,2)(3,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15);(1,3,5,8)(2,7,6,4);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72306510, #41×B(4,𝔽2), #72313082, … |
| maximal normal subgroup of | #72306510, #41×B(4,𝔽2), #72313082, … |
| maximal quotient of | #72311781 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524674);