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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,12)(14,15)(17,18,21,22)(19,25,26,23)(20,24,28,29)(27,31,32,30)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(2,5)(6,8)(9,10,13,16)(11,14,12,15)(18,22)(19,26)(20,28)(30,31)(35,42)(37,45)(39,41)(40,44);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,29)(21,22)(24,28)(25,26)(27,31)(30,32)(33,35,38,42)(34,39,36,41)(37,43,45,48)(40,46,44,47);(9,11,13,12)(10,14,16,15)(17,19,21,26)(18,23,22,25)(20,27,28,32)(24,30,29,31)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,36,44)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fg,d^2=fgi,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, 5209369);