Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,7)(9,10,14,12)(11,15,13,16)(18,22)(23,24)(25,26)(27,30)(28,31)(29,34)(32,38)(33,36)(35,40)(37,39);(1,2,4,5)(3,6,8,7)(10,12)(15,16)(17,18,21,22)(19,23,20,24)(26,27)(31,33)(32,34)(39,40);(2,5)(6,7)(9,11,14,13)(10,15,12,16)(17,19,21,20)(18,23,22,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,12,14,10)(11,15,13,16)(17,18)(19,24)(20,23)(21,22)(25,28)(26,31)(27,33)(29,37)(30,36)(32,40)(34,39)(35,38);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,16,12,15)(17,20,21,19)(18,24,22,23)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fgh,d^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2654358);