Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #3483 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,12,25,27,13,18,28,34,29,26,17,33)(11,20,23,36,24,37,35,40,21,16,30,32,38,22,31,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,21)(14,23)(15,24)(17,31)(18,20)(19,32)(25,35)(26,37)(27,39)(28,30)(29,38)(33,40)(34,36);(3,8)(5,7)(9,11)(10,16)(12,22)(13,21)(14,23)(15,24)(17,31)(18,20)(19,32)(25,35)(26,37)(27,39)(28,30)(29,38)(33,40)(34,36);(2,6)(5,7)(11,21)(16,20)(22,37)(23,30)(24,38)(31,35)(32,36)(39,40);(9,12,14,27,15,18,25,34,13,26,28,33,29,10,17,19)(11,22,23,39,24,20,35,36,21,37,30,40,38,16,31,32)Polycyclic presentation:
2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=dfi,e^a=ei,c^b=cfi,d^b=di,d^c=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72287764, #72287804, #72320653, … |
| maximal normal subgroup of | #72287764, #72287804, #72320653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529971);