Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140211 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,15,13,22,27,21,12,17,9,16,10,7,18,20)(8,19,24,34,29,14,30,32,26,35,28,23,25,33,36,31);(1,2,4,3)(5,7,12,22)(6,13,17,10)(8,23,26,34)(9,16,11,15)(14,30,33,36)(18,21,27,20)(19,28,35,24)(25,31,29,32);(1,3,4,2)(5,8,9,24,13,29,18,30,12,26,11,28,10,25,27,36)(6,14,15,31,22,35,21,34,17,33,16,32,7,19,20,23)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250430, #72250435, #72250464, … |
| maximal normal subgroup of | #72250430, #72250435, #72250464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56282);