Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| 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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1577 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,19)(13,20)(14,28)(15,32)(16,22)(18,24)(21,30)(23,34)(25,36)(26,38)(27,39)(29,35)(31,33)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11,16,12)(10,17,22,19)(13,25,15,23)(14,24,31,26)(18,33,38,28)(20,36,32,34)(21,35,39,37)(27,40,30,29);(9,11,16,12)(10,18,22,38)(13,23,15,25)(14,29,31,40)(17,33,19,28)(20,35,32,37)(21,36,39,34)(24,27,26,30);(3,8)(5,7)(9,12,16,11)(10,19,22,17)(13,23,15,25)(14,26,31,24)(18,28,38,33)(20,34,32,36)(21,37,39,35)(27,29,30,40);(2,6)(5,7)(9,13,16,15)(10,20,22,32)(11,23,12,25)(14,27,31,30)(17,34,19,36)(18,35,38,37)(21,28,39,33)(24,29,26,40)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72315208, #72315916, #72315921, … |
| maximal normal subgroup of | #72315208, #72315916, #72315921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531405);