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 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,27,17)(7,18,8,23,21,32,22,35)(9,25,28,14,29,33,36,34)(12,24,13,30,19,26,31,20);(1,2)(3,4)(5,7)(6,12)(8,10)(9,26)(11,21)(13,15)(14,23)(16,19)(17,31)(18,25)(20,28)(22,27)(24,29)(30,36)(32,33)(34,35);(2,4)(5,6)(7,19)(8,13)(9,14)(10,17)(11,16)(12,21)(15,27)(18,30)(20,32)(22,31)(23,24)(25,28)(26,35)(29,34)(33,36);(5,8,11,22)(6,13,16,31)(7,10,21,27)(9,20,29,30)(12,15,19,17)(14,32,34,18)(23,33,35,25)(24,36,26,28)Polycyclic presentation:
2^9|a^2=f,d^2=i,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, 408441);