Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 | |
1516 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)(11,15)(12,23)(13,18)(14,28)(16,20)(17,25)(19,30)(21,37)(22,31)(24,39)(26,38)(27,36)(29,33)(32,35)(34,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,18)(17,32)(19,30)(23,37)(24,34)(25,35)(26,33)(27,36)(28,31)(29,38)(39,40);(9,12,16,26)(10,17,20,34)(11,21,18,33)(13,24,15,32)(14,29,31,37)(19,35,36,39)(22,38,28,23)(25,27,40,30);(3,8)(5,7)(9,10)(11,13)(12,17)(14,19)(15,18)(16,20)(21,24)(22,27)(23,25)(26,34)(28,30)(29,35)(31,36)(32,33)(37,39)(38,40);(2,6)(5,7)(9,13,16,15)(10,18,20,11)(12,24,26,32)(14,27,31,30)(17,33,34,21)(19,28,36,22)(23,39,38,35)(25,29,40,37)Polycyclic presentation:
2^9|c^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=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531429);