Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(11,15)(14,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(34,38)(35,41)(36,44)(46,48);(10,13)(14,16)(19,25)(22,29)(26,31)(30,32)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,46,47,48);(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1526204);