Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C42 |
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×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5506 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)(7,10)(8,16)(9,12)(11,17)(13,15)(14,19)(18,20);(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,18)(16,17);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215866, #72215953, #72219491, … |
| maximal normal subgroup of | #72215866, #72215953, #72219491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350546);