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 | |
1296 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)(3,8)(4,7)(5,6)(11,15)(13,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34,39,42)(35,40,36,37)(38,41,46,48)(43,47,44,45);(2,6)(3,4)(19,25)(22,29)(26,31)(30,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,3,5,4)(2,7,6,8)(9,10,12,14)(11,13,15,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,2)(3,8)(4,7)(5,6)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=hi,d^2=hi,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, 5246925);