Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,14,11,16,10,27)(7,17,20,12,21,30,18,31)(9,23,22,32,26,13,25,15)(19,33,35,28,24,29,34,36);(1,2,3,4)(5,7)(6,12,16,31)(8,18)(9,24)(10,20)(11,21)(13,28,23,36)(14,17,27,30)(15,33,32,29)(19,26)(22,35)(25,34);(6,13,16,23)(8,10)(9,25)(12,28,31,36)(14,32,27,15)(17,29,30,33)(18,20)(19,35)(22,26)(24,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250667, #72250677, #72250681, … |
| maximal normal subgroup of | #72250667, #72250677, #72250681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54101);