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 | |
1314 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)(11,15)(12,16)(13,14)(18,22)(19,25)(24,29)(27,31)(34,39)(36,40)(37,45)(42,46);(10,14)(11,16)(20,28)(24,29)(27,31)(30,32)(33,34,38,39)(35,36,43,40)(37,41,45,47)(42,44,46,48);(11,16)(12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,43)(34,40,39,36)(37,46,45,42)(41,44,47,48);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,40)(34,35,39,43)(37,44,45,48)(41,42,47,46);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,16,14,11)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=f,e^2=fgi,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, 5071619);