Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141926 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1993 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,20,34)(8,22)(9,15)(10,27)(11,16)(12,18,30,36)(13,19,31,21)(14,25)(23,29,35,32)(24,28)(26,33);(5,7)(6,12,16,30)(8,23)(9,21)(10,19)(11,20)(13,14,31,28)(15,32,27,29)(17,33,34,22)(18,25)(24,36)(26,35);(2,4)(5,7)(6,13,16,31)(8,18)(9,19)(10,21)(11,20)(12,28,30,14)(15,17,27,34)(22,29,33,32)(23,25)(24,35)(26,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231475, #72231488, #72231535, … |
| maximal normal subgroup of | #72231475, #72231488, #72231535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56661);