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)(3,7)(4,6)(5,8)(11,15)(13,16)(17,18,21,22)(19,23,25,29)(20,24,27,28)(26,30,31,32)(36,43)(39,46)(41,47)(44,48);(1,2)(3,7)(4,6)(5,8)(18,22)(19,25)(20,27)(30,32)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,3,4,8)(2,5,6,7)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fgi,d^2=ghi,e^2=fh,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, 5244208);