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 | C22⋊#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16)(8,20,23,34)(9,24)(10,14)(11,15)(12,18)(13,22,29,21)(17,26,32,36)(19,27)(25,28)(30,31,33,35);(1,2,3,4)(5,7)(6,12)(8,17)(9,25)(10,18)(11,19)(13,26)(14,27)(15,28)(16,24)(20,33)(21,35)(22,31)(23,32)(29,36)(30,34);(5,8)(6,13,15,29)(7,17)(9,21)(10,22)(11,23)(12,26,28,36)(14,20,24,34)(16,30,27,33)(18,31)(19,32)(25,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^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 | #72210904, #72210945, #72210949, … |
| maximal normal subgroup of | #72210904, #72210945, #72210949, … |
| maximal quotient of | #72210860, #72211172, #72280379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30653);