Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
958 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)(7,18,22,13)(8,25,26,32)(9,27)(10,15)(11,17)(12,20,31,21)(14,28)(16,30)(19,35)(23,36,24,33)(29,34);(2,4)(5,7,10,23,11,22,30,24)(6,12,15,25,17,31,16,32)(8,19,20,9,26,28,21,29)(13,14,33,34,18,35,36,27);(5,8)(6,13)(7,19)(9,23)(10,20)(11,26)(12,14)(15,33)(16,36)(17,18)(21,30)(22,28)(24,29)(25,34)(27,32)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72314250 |
| maximal normal subgroup of | #72314250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53983);