Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| 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 | |
476 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)(5,6,12,17,13,18,30,19,14,20,9,29,10,31,11,32)(7,21,24,16,25,34,8,27,26,36,22,35,23,33,28,15);(5,7)(6,15,20,27)(8,12)(9,28)(10,25)(11,22)(13,23)(14,26)(16,18,35,31)(17,34,29,33)(19,21,32,36)(24,30);(5,8,9,7,13,22,11,25,14,28,12,26,10,24,30,23)(6,16,17,34,18,27,19,36,20,35,29,33,31,15,32,21)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,g^b=gh,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250642, #72250657, #72250681, … |
| maximal normal subgroup of | #72250642, #72250657, #72250681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56289);