Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,18,28)(8,19,20,11)(9,22,24,13)(10,14)(12,17)(16,30)(21,26)(23,27)(25,32,33,36)(29,31)(34,35);(1,2,3,4)(6,11,14,19)(8,20)(12,25,27,33)(13,26,22,34)(15,29,28,30)(16,31)(21,35)(32,36);(5,7,10,18)(6,12,14,27)(8,16,20,31)(9,23,24,17)(11,25,19,33)(13,28,22,15)(21,36,35,32)(26,30,34,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=di,f^b=fh,e^c=ei,f^c=fhi,f^d=fi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53895);