Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1853 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,10,16,12,17,11,29)(7,18,20,32,22,35,21,33)(8,23,26,15,9,28,27,14)(13,24,30,19,31,25,34,36);(5,7)(6,13,17,31)(8,24)(9,25)(10,21)(11,20)(12,22)(14,32,15,33)(16,34,29,30)(18,28,35,23)(19,27)(26,36);(1,3,4,2)(5,7)(6,13,17,31)(8,25)(9,24)(10,21)(11,20)(12,22)(14,33,15,32)(16,34,29,30)(18,23,35,28)(19,26)(27,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250650, #72250659, #72250661, … |
| maximal normal subgroup of | #72250650, #72250659, #72250661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54487);