Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136733 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(7,16,19,18)(8,21)(9,25)(10,24)(12,17,30,35)(13,22)(14,28)(20,23,29,27)(26,33)(31,32,34,36);(1,2,4,3)(5,6)(7,17)(8,22)(9,26)(10,28)(11,15)(12,18)(13,21)(14,24)(16,30)(19,35)(20,32)(23,34)(25,33)(27,31)(29,36);(1,3,4,2)(5,7)(6,12)(8,23)(9,27)(10,29)(11,19)(13,32)(14,34)(15,30)(16,25)(17,26)(18,21)(20,24)(22,35)(28,31)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250446, #72250476, #72250485, … |
| maximal normal subgroup of | #72250446, #72250476, #72250485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55953);