Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501248 |
| 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 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,12,18,11,30)(7,19,21,34,22,14,8,23)(9,25,27,35,29,31,28,16)(13,26,32,36,33,24,15,20);(1,2)(3,4)(5,7)(6,13)(8,11)(9,26)(10,21)(12,22)(14,31)(15,30)(16,23)(17,32)(18,33)(19,25)(20,28)(24,29)(27,36)(34,35);(2,4)(5,8)(6,14)(7,11)(9,24)(10,22)(12,21)(13,31)(15,16)(17,34)(18,19)(20,28)(23,30)(25,33)(26,29)(27,36)(32,35);(5,8)(6,15)(7,10)(9,20)(11,22)(12,21)(13,17)(14,16)(18,32)(19,35)(23,25)(24,28)(26,27)(29,36)(30,33)(31,34)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,d^b=di,e^b=eh,h^b=hi,f^c=fi,e^d=eh,h^d=hi56 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, 408444);