Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298939 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,9)(7,10)(11,12)(14,19)(15,20)(16,17);(1,2)(3,4)(5,6,10,12,8,11,7,9)(13,14,17,15,18,20,16,19);(2,4)(13,15,17,20,18,19,16,14);(5,6,10,12,8,11,7,9)(13,14,16,19,18,20,17,15)Polycyclic presentation:
2^9|b^2=eh,c^2=f,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214806 |
| maximal normal subgroup of | #72214806 |
| maximal quotient of | #72214497 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 206209);