Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,13)(14,15)(19,23)(21,24)(26,30)(27,33)(28,35)(38,40);(1,2)(3,5)(4,7)(6,8)(10,15)(11,13)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,37)(34,39);(2,3)(6,7)(9,11,14,13)(10,16,15,12)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,11,15,13)(17,19,20,23)(18,21,22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,16,15,12)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=fg,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419635);