Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1730 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)(8,22,24,29)(9,15)(10,14)(11,16)(12,19)(13,23,30,25)(18,32,35,26)(20,28)(21,27)(31,34,36,33);(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,25,33,24,35,23,34)(13,26,29,36,30,32,22,31);(2,4)(5,8)(6,13,16,30)(7,18)(9,23)(10,25)(11,24)(12,26,28,32)(14,29,15,22)(17,31,27,36)(19,33)(20,34)(21,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,d^b=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72210945, #72210958, #72210978, … |
| maximal normal subgroup of | #72210945, #72210958, #72210978, … |
| maximal quotient of | #72239213 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30654);