Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,19)(13,26)(14,30)(15,18)(17,22)(20,28)(21,32)(23,31)(24,34)(25,37)(27,33)(29,40)(35,38)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,24)(13,22)(14,21)(15,23)(17,26)(18,31)(19,34)(20,33)(25,39)(27,28)(29,38)(30,32)(35,40)(36,37);(1,4,5,2)(3,7,8,6)(9,11)(10,17)(12,21)(13,27)(14,24)(15,23)(16,26)(18,35)(19,33)(20,34)(22,28)(25,38)(29,39)(30,36)(31,40)(32,37);(3,8)(6,7)(10,18)(16,31)(17,35)(19,30)(20,37)(26,40)(32,34)(33,36);(9,12,15,14)(10,19,18,30)(11,21,23,24)(13,25,29,28)(16,32,31,34)(17,33,35,36)(20,26,37,40)(22,27,38,39)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216688, #72224998, #72225002, … |
| maximal normal subgroup of | #72216688, #72224998, #72225002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531022);