Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501244 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,16,10,27)(7,17,19,32,20,31,26,13)(8,21,23,33,25,35,24,14)(12,22,28,34,30,36,29,18);(1,2)(3,4)(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,14)(15,28)(16,30)(17,21)(18,24)(23,34)(25,36)(27,29)(31,35)(32,33);(2,4)(5,7)(6,13)(8,18)(9,26)(10,19)(11,20)(12,14)(15,31)(16,32)(17,27)(21,29)(22,24)(23,36)(25,34)(28,35)(30,33);(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,14)(15,28)(16,30)(17,21)(18,24)(23,34)(25,36)(27,29)(31,35)(32,33)Polycyclic presentation:
2^9|a^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,f^c=fi,e^d=eh,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72323008 |
| maximal normal subgroup of | #72323008 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408440);