Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,12)(13,16)(14,15)(17,18,21,19)(20,23,24,22)(25,26,30,33)(27,31,28,35)(29,32,37,40)(34,38,36,39);(1,3)(2,6)(4,7)(5,8)(10,15)(11,13)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(1,2,5,7)(3,6,8,4)(9,11,14,13)(10,16,15,12)(17,19,21,18)(20,23,24,22)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,40,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20,21,24)(18,22,19,23)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(1,4,5,6)(2,3,7,8)(9,13,14,11)(10,16,15,12)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5787579);