Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C23 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ 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)(5,6,9,15,11,14,10,16)(7,12,19,26,18,25,23,30)(8,13,21,27,20,28,22,29)(17,31,33,36,32,24,34,35);(5,7)(6,12,14,25)(8,17)(9,23)(10,19)(11,18)(13,24,28,31)(15,30,16,26)(20,32)(21,34)(22,33)(27,36,29,35);(2,4)(5,8,10,22,11,20,9,21)(6,13,16,29,14,28,15,27)(7,17,19,33,18,32,23,34)(12,24,26,35,25,31,30,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,e^b=ei,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237651, #72237657, #72237664, … |
| maximal normal subgroup of | #72237651, #72237657, #72237664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30694);