Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1734 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)(5,6,9,15,11,16,10,14)(7,17,19,12,20,27,21,26)(8,13,23,28,24,29,22,30)(18,32,35,36,34,31,33,25);(5,7)(6,12,16,26)(8,18)(9,19)(10,21)(11,20)(13,25,29,36)(14,17,15,27)(22,33)(23,35)(24,34)(28,32,30,31);(5,8,10,22,11,24,9,23)(6,13,14,30,16,29,15,28)(7,18,21,33,20,34,19,35)(12,25,17,31,26,36,27,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237182, #72237183, #72237188, … |
| maximal normal subgroup of | #72237182, #72237183, #72237188, … |
| maximal quotient of | #72239105, #72239223, #72239224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30682);