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 | |
1308 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)(3,5)(4,7)(6,8)(11,15)(13,16)(18,22)(23,28)(24,29)(30,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(19,25)(23,28)(26,31)(30,32)(35,42)(36,44)(38,41)(39,46);(1,2,5,3)(4,6,8,7)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,38)(36,43)(37,42)(39,45)(40,41)(44,48)(46,47);(1,3,5,2)(4,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,37,44)(34,39,40,46)(35,43,42,48)(38,45,41,47);(1,4)(2,6)(3,7)(5,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(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=fi,c^2=f,d^2=g,e^2=gh,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, 4781355);