Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,12)(8,13)(9,17)(10,14)(11,15)(16,19)(18,20);(5,7)(6,11)(8,15)(9,18)(10,16)(12,17)(13,19)(14,20);(1,2,3,4)(5,7)(6,12)(8,15)(9,18)(10,16)(11,13)(14,19)(17,20);(5,7)(6,12)(8,15)(9,18)(10,16)(11,13)(14,19)(17,20);(5,8,10,9)(6,13,14,17)(7,15,16,18)(11,19,20,12)Polycyclic presentation:
2^9|c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215857, #72215866, #72219482, … |
| maximal normal subgroup of | #72215857, #72215866, #72219482, … |
| maximal quotient of | C4×#24222, #72306301, #72306311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349751);