Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| 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 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,8,14,11,12,10,16)(7,17,20,13,21,30,19,27)(9,23,22,33,26,15,24,31)(18,32,25,28,36,34,35,29);(6,12)(7,18,21,36)(8,10)(9,24)(13,28,27,29)(15,31)(17,34,30,32)(19,25,20,35)(22,26)(23,33);(5,7)(6,13,12,27)(8,19)(9,25)(10,20)(11,21)(14,17,16,30)(15,32,23,34)(18,22)(24,36)(26,35)(28,31,29,33)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239861, #72240846, #72240855, … |
| maximal normal subgroup of | #72239861, #72240846, #72240855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56157);