Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,22)(19,23,27,25)(20,24,29,30)(26,31,28,32)(33,34)(35,41)(36,38)(37,44)(39,42)(40,45)(43,47)(46,48);(3,7)(6,8)(9,11,13,10)(12,15,16,14)(18,22)(23,25)(24,30)(31,32)(34,36)(35,42)(37,45)(47,48);(1,2,4,5)(3,6,7,8)(17,19,21,27)(18,23,22,25)(20,28,29,26)(24,32,30,31)(33,35)(34,39)(36,41)(37,46)(38,42)(40,48)(43,45)(44,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,48,46,47);(1,3,4,7)(2,6,5,8)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,47)(41,48)(42,46)Polycyclic presentation:
2^9|a^2=gi,b^2=fghi,c^2=fg,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4976369);