Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11)(8,18)(9,14)(10,15)(12,17)(13,19)(16,20)(21,22,25,28)(23,26,30,34)(24,27,32,35)(29,36,31,33);(1,2,4,3)(5,7,10,17)(6,11,15,12)(8,19,16,9)(13,20,14,18)(21,23)(22,26)(24,31)(25,30)(27,33)(28,34)(29,32)(35,36);(5,8,19,7,10,16,9,17)(6,12,14,20,15,11,13,18)(21,24)(22,27)(23,29,30,31)(25,32)(26,33,34,36)(28,35);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=fh,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 204563);