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 | |
1452 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)(10,13)(11,15)(17,18,21,20)(19,22,23,24)(25,26,30,31)(27,29,33,39)(28,35,37,32)(34,36,40,38);(1,2,5,3)(4,7,8,6)(11,15)(14,16)(19,23)(22,24)(26,31)(27,33)(32,35)(34,40);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,19,21,23)(18,22,20,24)(25,27)(26,29)(28,34)(30,33)(31,39)(32,38)(35,36)(37,40);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,28,30,37)(26,32,31,35)(27,34,33,40)(29,38,39,36);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)(25,29)(26,33)(27,31)(28,36)(30,39)(32,34)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=fgh,d^2=h,e^2=fgh,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, 4095535);