Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11,15,8)(9,12)(10,13)(14,16,17,19)(18,20);(1,3,4,2)(5,7)(6,11,13,8)(9,17,18,14)(10,15)(12,19)(16,20);(2,3)(5,8,10,11)(6,7)(9,16,18,19)(12,14)(13,15)(17,20);(5,9)(6,12)(7,14)(8,16)(10,18)(11,19)(13,20)(15,17)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=ei,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225672, #72225674, #72225684, … |
| maximal normal subgroup of | #72225672, #72225674, #72225684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340096);