Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501868 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14)(7,15,17,35)(8,19,21,36)(9,25,23,13)(11,27,28,16)(12,30,32,24)(18,31,26,33)(20,29,22,34);(1,2)(3,4)(5,7,9,26,10,17,23,18)(6,11,13,34,14,28,25,29)(8,20,24,27,21,22,30,16)(12,31,19,35,32,33,36,15);(2,4)(8,21)(12,32)(15,35)(16,27)(19,36)(20,22)(24,30)(31,33);(5,7)(6,11)(8,22)(9,26)(10,17)(12,33)(13,34)(14,28)(15,19)(16,24)(18,23)(20,21)(25,29)(27,30)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=i,b^2=fh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,e^d=eh,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72321029 |
| maximal normal subgroup of | #72321029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409066);