Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(15,16)(19,20);(2,4)(13,14)(15,19)(16,18)(17,20);(6,9)(10,12)(14,18)(15,16)(19,20);(14,18)(15,16)(19,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225125, #72257831, #72305390, … |
| maximal normal subgroup of | #72225125, #72257831, #72305390, … |
| maximal quotient of | #72225017, #72264308, #72264574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361093);