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 | |
4847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(11,18)(19,20);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,19)(16,20)(17,18);(2,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,20)(14,18);(1,2)(3,4)(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,19,18,20);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,20,18,19)Polycyclic presentation:
2^9|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, C2×#6573711, #72268318, … |
| maximal normal subgroup of | D4×#3025, C2×#6573711, #72268318, … |
| maximal quotient of | C2×#6573711, #72268318, #72268320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480907);