Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1656 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,13)(14,16)(19,23)(22,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(2,3)(4,8)(9,10,12,13)(11,15,16,14)(17,18,21,20)(19,22,23,24)(26,31)(27,33)(29,37)(36,39);(2,3)(6,7)(10,13)(14,15)(17,18,21,20)(19,22,23,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,24,23,22)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=ghi,d^2=gh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1545050);