Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,8,7)(9,10,13,16)(11,14,12,15)(19,25)(20,27)(23,28)(24,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,5)(6,8)(18,22)(23,28)(24,29)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,4)(6,7)(9,11,13,12)(10,14,16,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=i,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5373894);