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 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,4)(6,7)(9,10)(11,16)(12,15)(13,14)(20,27)(24,29)(25,31)(30,32)(33,34,37,38)(35,41,42,39)(36,44,46,40)(43,47,45,48);(1,2,5,4)(3,7,8,6)(9,11)(10,12)(13,16)(14,15)(18,22)(23,28)(24,29)(30,32)(34,38)(35,42)(40,44)(43,45);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,42)(34,39,38,41)(36,45,46,43)(40,48,44,47);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,16,15,12)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,48)(42,45)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fh,d^2=fgh,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, 4780258);