Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,12)(8,15)(9,18)(10,13)(11,14)(16,19)(17,20);(2,3)(5,7,11,16)(6,12,14,19)(8,9,17,10)(13,15,18,20);(1,2,4,3)(5,8,9,7,11,17,10,16)(6,12,13,15,14,19,18,20);(1,3,4,2)(5,6)(7,15)(8,12)(9,13)(10,18)(11,14)(16,20)(17,19);(7,16)(8,17)(12,19)(15,20)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72213663, #72213751, … |
| maximal normal subgroup of | #72213480, #72213663, #72213751, … |
| maximal quotient of | #72226917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351507);