Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1298 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)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(19,25)(23,28)(26,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(3,8)(5,7)(10,12)(11,15)(18,22)(19,25)(24,29)(26,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,12,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,16,15,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5910301);