Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196299 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35274 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,10,13,11,14,9,12)(7,8,15,17,16,19,20,18);(1,2,4,3)(5,7,10,15,11,16,9,20)(6,8,13,17,14,19,12,18);(5,8)(6,7,14,16)(9,17)(10,18)(11,19)(12,15,13,20);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227223, #72227280, D4×#1910, … |
| maximal normal subgroup of | #72227223, #72227280, D4×#1910, … |
| maximal quotient of | #72258195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276238);