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 | |
1356 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)(3,8)(4,5)(6,7)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,29)(26,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3)(2,5)(4,6)(7,8)(10,12)(11,15)(19,25)(23,28)(26,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,7)(3,4)(9,10,13,12)(11,16,15,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4,6,3)(2,8,7,5)(9,11,13,15)(10,14,12,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,6,8)(2,3,7,4)(9,12,13,10)(11,16,15,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=gi,d^2=g,e^2=fgh,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, 4401903);