Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437473 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,11)(7,15)(8,9)(13,14)(16,17)(18,19);(1,2)(3,4)(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,20)(17,18);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20);(2,4);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,19,20,18)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72215995, #72216013, … |
| maximal normal subgroup of | #72214076, #72215995, #72216013, … |
| maximal quotient of | C2×#6437475, #72236700, #72239576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344671);