Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,20,25,33)(8,26,22,35)(9,16)(10,30)(11,31)(12,32)(13,19)(14,23,34,24)(15,28,36,27)(17,21)(18,29);(1,2,3,4)(5,7,21,28,9,22,10,24,13,25,29,27,12,8,11,23)(6,14,30,26,32,36,17,33,19,34,31,35,16,15,18,20);(5,8,11,28,12,25,29,23,13,22,10,27,9,7,21,24)(6,15,18,26,16,34,31,20,19,36,17,35,32,14,30,33);(5,9,13,12)(6,16,19,32)(7,22,25,8)(10,29,11,21)(14,15,34,36)(17,30,18,31)(20,35,33,26)(23,28,24,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224815, #72224882, #72224883, … |
| maximal normal subgroup of | #72224815, #72224882, #72224883, … |
| maximal quotient of | #72225237, #72225275, #72225299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402436);