Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(10,13)(11,15)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34)(35,36)(37,45)(38,39)(40,44)(41,47)(42,46)(43,48);(1,2,5,4)(3,6,7,8)(11,15)(14,16)(19,25)(23,26)(27,31)(30,32)(34,39)(35,40)(41,45)(42,48);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,38,40)(34,36,39,44)(37,42,47,48)(41,43,45,46);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,47)(45,48);(1,4,5,2)(3,8,7,6)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,47)(34,41,39,45)(35,42,40,48)(36,43,44,46)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4825985);