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 | |
2090 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)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(25,26,30,33)(27,34,32,29)(28,36,38,31)(35,39,40,37);(6,8)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27,30,32)(26,29,33,34)(28,35,38,40)(31,39,36,37);(5,6)(7,8)(9,12,15,11)(10,16,13,18)(14,19,22,20)(17,24,21,23)(25,27)(26,29)(28,35)(30,32)(31,39)(33,34)(36,37)(38,40);(1,2,3,4)(9,13,15,10)(11,16,12,18)(14,17,22,21)(19,24,20,23)(25,28,30,38)(26,31,33,36)(27,35,32,40)(29,39,34,37);(9,14,15,22)(10,17,13,21)(11,19,12,20)(16,23,18,24)(25,29)(26,32)(27,33)(28,37)(30,34)(31,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=i,e^2=fg,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, 3371375);