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) = C25×C4 |
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 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,15,16,14)(20,27)(24,29)(25,31)(30,32)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(10,12)(11,16)(18,22)(23,28)(24,29)(30,32)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(10,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=h,e^2=i,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, 5072306);