Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1400 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)(3,7)(11,15)(13,16)(17,18,22,25)(19,20,26,23)(21,29,30,24)(27,31,32,28)(34,35)(36,43)(40,42)(44,48);(3,7)(6,8)(19,26)(20,23)(27,32)(28,31)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,29)(30,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(17,21,22,30)(18,24,25,29)(19,27,26,32)(20,28,23,31)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431618);