Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,27)(22,23)(24,28)(25,31)(26,32)(29,30)(34,35)(36,43)(37,45)(46,47);(3,7)(6,8)(11,15)(13,16)(18,23)(19,25)(20,28)(30,32)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,19,22,25)(18,21,23,31)(20,29,28,26)(24,32,27,30)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,27)(25,29)(31,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21)(18,19)(20,30)(22,31)(23,25)(24,26)(27,29)(28,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fghi,d^2=fh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5363194);