Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62832 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3484 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,8,12,6)(5,9)(10,15)(11,14,16,13);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(3,10)(11,16);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216336, #72216366, #72233535, … |
| maximal normal subgroup of | #72216336, #72216366, #72233535, … |
| maximal quotient of | #72233187, #72233190, #72233517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400676);