Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1318 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,47,46,48);(1,3)(2,7)(4,5)(6,8)(9,11,13,12)(10,14,16,15)(18,22)(19,26)(24,28)(27,32)(34,38)(35,42)(36,45)(47,48);(2,4)(3,8)(9,11,13,12)(10,14,16,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,35,37,42)(34,39,38,41)(36,46,45,43)(40,48,44,47);(1,4,6,2)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,15,16,14)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36,37,45)(34,40,38,44)(35,43,42,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=fgi,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5116680);