Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C24 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
556 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)(5,6,9,15,11,16,27,33,12,17,26,14,10,29,8,24)(7,18,21,34,23,35,25,13,19,31,28,32,22,36,20,30);(2,4)(5,7)(6,13,16,32,17,30,29,34)(8,25)(9,28)(10,22)(11,23)(12,19)(14,31,24,36,15,18,33,35)(20,27)(21,26);(7,19)(13,30)(18,31)(20,25)(21,28)(22,23)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ci,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210536, #72224803, #72224805, … |
| maximal normal subgroup of | #72210536, #72224803, #72224805, … |
| maximal quotient of | #72210328, #72323174 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33989);