Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136643 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,16,20,34)(8,22)(9,14)(10,27)(11,15)(12,18,28,35)(13,25)(17,23,31,36)(19,33,21,29)(24,30)(26,32);(1,2,3,4)(5,6)(7,17)(8,13)(9,27)(10,14)(11,15)(12,19)(16,23)(18,33)(20,31)(21,28)(22,25)(24,32)(26,30)(29,35)(34,36);(5,7)(6,12)(8,23)(9,21)(10,19)(11,20)(13,29)(14,31)(15,28)(16,32)(17,27)(18,24)(22,34)(25,35)(26,36)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=ehi,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72224939, #72226819, … |
| maximal normal subgroup of | #72224903, #72224939, #72226819, … |
| maximal quotient of | #72224901, #72237295, C2×#148049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55245);