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 | |
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:
(2,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(9,10,13,16)(11,12,15,14)(18,19)(20,27)(21,29)(30,31)(36,44)(39,46)(42,48)(45,47);(11,15)(12,14)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,16,12)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=g,d^2=fh,e^2=fi,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, 5042565);