Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,17)(9,12)(10,13)(14,16)(15,19)(18,20);(5,6,9,17,10,13,8,12)(7,14,18,19,15,20,16,11);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,15)(12,20)(13,19)(14,17);(5,6)(7,11)(8,12)(9,17)(10,13)(14,18)(15,19)(16,20);(7,15)(11,19)(14,20)(16,18)Polycyclic presentation:
2^9|b^2=fi,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213250, #72214082, #72215953, … |
| maximal normal subgroup of | #72213250, #72214082, #72215953, … |
| maximal quotient of | #72226283, C2×#6443319 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350515);