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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(9,10,15,13)(11,16,14,12)(17,18,21,22)(19,25,26,23)(20,28,30,24)(27,31,29,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,3)(6,7)(9,11,15,14)(10,16,13,12)(18,22)(19,26)(24,28)(27,29)(34,38)(39,41)(40,44)(47,48);(1,2,5,3)(4,6,8,7)(9,12)(10,14)(11,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,16,14,12)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,5,8)(2,6,3,7)(9,14,15,11)(10,16,13,12)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=fi,d^2=h,e^2=fg,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, 4796377);