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) = 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 | |
1294 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)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,31)(26,27)(28,30)(29,32)(33,34,37,38)(35,39,41,42)(36,44,46,40)(43,48,45,47);(9,11,14,16)(10,12,13,15)(18,20)(23,27)(24,28)(31,32)(34,38)(35,41)(40,44)(43,45);(10,13)(11,16)(17,19,22,26)(18,23,20,27)(21,29,30,25)(24,32,28,31)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,46)(44,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20,22,18)(19,23,26,27)(21,28,30,24)(25,31,29,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,12,16,15)(17,21,22,30)(18,24,20,28)(19,25,26,29)(23,31,27,32)(33,36,37,46)(34,40,38,44)(35,43,41,45)(39,47,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fh,d^2=fh,e^2=i,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, 5044535);