Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(18,22)(19,25)(20,27)(30,32)(33,34,37,38)(35,39,43,41)(36,44,46,40)(42,48,45,47);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(19,25)(23,28)(26,31)(30,32)(34,38)(39,41)(40,44)(47,48);(1,2,4,5)(3,6,7,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,3,4,7)(2,6,5,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2648429);