Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2114 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)(9,10)(11,16)(12,21)(13,17)(14,15)(18,19)(20,23)(22,24)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(5,6,7,8)(10,15)(12,17)(16,23)(18,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(5,6)(7,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,2,3,4)(9,12)(10,13)(11,18)(14,17)(15,21)(16,22)(19,23)(20,24)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(9,13)(10,17)(11,19)(12,15)(14,21)(16,18)(20,22)(23,24)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3372417);