Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(8,16)(10,19)(12,17)(14,18);(1,2)(3,4)(5,6)(7,13)(8,17)(9,11)(10,14)(12,16)(15,20)(18,19);(1,3,4,2)(5,7)(6,10)(8,18)(9,15)(11,19)(12,20)(13,17)(14,16);(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,17)(18,20)Polycyclic presentation:
2^9|c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260544, #72260944, #72260961, … |
| maximal normal subgroup of | #72260544, #72260944, #72260961, … |
| maximal quotient of | #72260553 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301749);