Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,12,15,14)(17,18,22,21)(19,20,24,23)(25,26)(27,28)(29,37)(30,31)(32,33)(34,38)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,21)(22,24)(26,31)(27,33)(29,38)(36,40);(9,11)(10,12)(13,15)(14,16)(17,20,22,23)(18,24,21,19)(25,27)(26,32)(28,31)(29,39)(30,33)(34,36)(35,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28,30,32)(26,33,31,27)(29,40,38,36)(34,35,37,39);(1,4,5,3)(2,7,8,6)(9,12)(10,15)(11,16)(13,14)(17,21,22,18)(19,20,24,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=gh,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4202130);