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 | |
1302 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)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,46)(42,48,45,47);(1,3)(2,7)(4,5)(6,8)(9,11,13,12)(10,14,16,15)(18,19)(20,27)(21,29)(30,31)(34,38)(36,44)(39,41)(42,45);(2,4)(3,8)(9,11,13,12)(10,14,16,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,44)(46,47);(1,4,6,2)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,15,16,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=gi,d^2=h,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, 5116670);