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) = C28 |
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 | |
2308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(6,8)(10,11)(15,17)(16,18)(23,24)(28,37)(31,36)(34,40)(38,39);(5,6)(7,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2)(3,4)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=f,e^2=fg,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, 4038874);