Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1924 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)(3,7)(17,18)(19,22)(20,24)(21,23)(25,26)(27,34)(28,32)(29,37)(30,31)(33,39)(35,36)(38,40);(10,13)(14,16)(17,19,21,24)(18,22,23,20)(26,31)(27,32)(33,37)(35,40);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,22)(23,24)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|b^2=fgi,c^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1309719);