Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2075 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,13,11,7,16,15)(8,19,23,31,22,34,20,35)(10,25,24,14,27,30,32,18)(12,26,29,21,17,33,28,36);(1,2)(3,4)(5,7)(6,11)(8,12)(9,13)(10,18)(14,27)(15,16)(17,22)(19,21)(20,29)(23,28)(24,30)(25,32)(26,31)(33,35)(34,36);(2,4)(5,8,9,20,11,22,16,23)(6,12,13,28,7,17,15,29)(10,26,24,36,27,33,32,21)(14,31,30,19,18,35,25,34)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,e^c=ehi,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239853, #72240847, #72240857, … |
| maximal normal subgroup of | #72239853, #72240847, #72240857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55816);