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 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,9)(7,11,15,14)(8,12,17,16)(13,19,20,18);(1,2)(3,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20);(2,4)(5,8)(6,12)(7,13)(9,16)(10,17)(11,18)(14,19)(15,20);(1,2)(3,4)(5,6,10,9)(7,11,15,14)(8,12,17,16)(13,18,20,19);(2,4)(5,9,10,6)(7,14,15,11)(8,16,17,12)(13,19,20,18)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,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 | D4×#3025, #72252415, #72252423, … |
| maximal normal subgroup of | D4×#3025, #72252415, #72252423, … |
| maximal quotient of | C2×#6573714, #72267129, #72267158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480910);