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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,12)(13,14)(15,16)(18,22)(20,27)(23,28)(26,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,48)(45,47);(1,3)(2,6)(4,8)(5,7)(10,14)(12,15)(18,22)(19,25)(24,29)(26,31)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,2)(3,4)(5,8)(6,7)(9,12,13,15)(10,11,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,4)(2,7)(3,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,40,41,47)(35,43,36,44)(39,46,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fgh,d^2=fi,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, 4802753);