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 |
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 | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,20,31)(8,13)(9,22)(10,25)(11,15)(12,19,27,34)(14,24)(17,29,18,30)(21,28)(23,35)(26,33,36,32);(2,4)(7,17,20,18)(9,23)(12,26,27,36)(14,28)(16,29,31,30)(19,33,34,32)(21,24)(22,35);(5,7,9,19,10,17,24,32,11,20,23,34,8,18,21,33)(6,12,14,16,13,26,22,30,15,27,28,31,25,36,35,29);(5,8,11,10)(6,13,15,25)(7,18,20,17)(9,21,23,24)(12,26,27,36)(14,22,28,35)(16,30,31,29)(19,33,34,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239492, #72250290, #72250303, … |
| maximal normal subgroup of | #72239492, #72250290, #72250303, … |
| maximal quotient of | #72285702, #72309629, #72310163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402239);