Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10,15,14)(11,12,16,13)(19,23)(20,24)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,39)(38,40);(2,3)(6,7)(9,11)(10,13)(12,14)(15,16)(17,18,21,22)(19,24,23,20)(26,27)(28,35)(32,34)(36,40);(1,2,5,3)(4,6,8,7)(9,12,15,13)(10,16,14,11)(17,19)(18,20)(21,23)(22,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,3,5,2)(4,7,8,6)(9,13,15,12)(10,16,14,11)(17,18)(19,20)(21,22)(23,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,10)(11,12,16,13)(17,20)(18,19)(21,24)(22,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637188);