Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136735 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| 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 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,18,12,20,22,19)(7,21,24,14,26,35,25,32)(8,27,23,13,29,33,28,34)(9,17,10,15,30,16,31,36);(1,2)(3,4)(5,7)(6,13)(8,9)(10,23)(11,24)(12,26)(14,15)(16,35)(17,21)(18,33)(19,27)(20,34)(22,25)(28,31)(29,30)(32,36);(2,4)(5,8,9,7,22,23,31,24,12,29,30,26,11,28,10,25)(6,14,15,13,18,21,16,27,20,32,36,34,19,35,17,33)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321658, #72321689, #72321723, … |
| maximal normal subgroup of | #72321658, #72321689, #72321723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55703);