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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(11,12)(15,16)(18,22)(19,25)(24,29)(27,31)(34,38)(35,41)(40,44)(43,45);(10,14)(15,16)(20,28)(24,29)(27,31)(30,32)(33,34,37,38)(35,42,41,39)(36,44,46,40)(43,47,45,48);(9,10,13,14)(11,15,12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,37,41)(34,39,38,42)(36,45,46,43)(40,48,44,47);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=h,d^2=fh,e^2=fi,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, 5296022);