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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,12)(15,16)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31)(33,34)(35,39)(36,38)(37,44)(40,43)(41,46)(42,48)(45,47);(10,14)(15,16)(18,22)(19,26)(20,28)(31,32)(33,35,38,43)(34,39,36,40)(37,45,46,42)(41,48,44,47);(9,10)(11,15)(12,16)(13,14)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,36,38,34)(35,40,43,39)(37,44,46,41)(42,48,45,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,24,22,30)(19,27,26,29)(23,31,25,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,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, 5258682);