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 | |
1358 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)(9,10,14,12)(11,15,13,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(9,11,14,13)(10,15,12,16)(18,19)(20,27)(21,29)(30,31)(35,42)(36,44)(38,41)(39,46);(1,2,5,4)(3,6,7,8)(9,12,14,10)(11,15,13,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35,37,42)(34,38,40,41)(36,43,44,48)(39,45,46,47);(1,3)(2,6)(4,8)(5,7)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fhi,d^2=f,e^2=fghi,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, 4496441);