Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1568 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)(4,5)(6,8)(9,10,14,12)(11,15,13,16)(19,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,13)(10,15,12,16)(18,22)(23,24)(26,31)(28,32)(34,39)(36,40);(2,5)(6,7)(9,12,14,10)(11,15,13,16)(17,18,21,22)(19,23,20,24)(25,27)(26,32)(28,31)(29,37)(30,33)(34,36)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(17,19,21,20)(18,23,22,24)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,15,12,16)(17,20,21,19)(18,24,22,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5026659);