Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(3,7)(11,12)(14,15)(17,18,21,20)(19,22,23,24)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,40)(38,39);(3,7)(6,8)(9,10,13,16)(11,14,12,15)(19,23)(22,24)(26,31)(27,33)(34,39)(35,40);(1,2,5,4)(3,6,7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,37)(36,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fg,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,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, 2675276);