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 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1264 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)(9,10,14,17)(11,15,20,28)(12,16,25,31)(13,26,24,19)(18,33,32,23)(21,27,34,38)(22,37,35,30)(29,40,39,36);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,28)(18,29)(19,30)(23,36)(24,35)(25,34)(26,37)(31,38)(32,39)(33,40);(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(21,36)(22,35)(25,33)(26,32)(28,38)(29,30)(34,40)(37,39);(1,4,5,2)(3,6,8,7)(10,17)(15,28)(16,31)(18,32)(19,26)(27,38)(29,39)(30,37);(11,20)(15,28)(21,34)(22,35)(27,38)(29,39)(30,37)(36,40)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=f,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 | #72216687, #72224999, C2×#7623937, … |
| maximal normal subgroup of | #72216687, #72224999, C2×#7623937, … |
| maximal quotient of | C2×#7623937, #72319556, #72321852 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531133);