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 | |
1316 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)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,22)(21,28)(24,30)(25,27)(26,32)(29,31)(35,41)(36,42)(37,45)(39,46);(11,15)(13,16)(18,20)(19,25)(24,28)(26,29)(33,34,38,40)(35,42,41,36)(37,46,45,39)(43,44,47,48);(9,11,12,15)(10,13,14,16)(17,19,22,25)(18,23,20,27)(21,29,30,26)(24,32,28,31)(33,35,38,41)(34,36,40,42)(37,43,45,47)(39,44,46,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,36)(34,35)(37,44)(38,42)(39,43)(40,41)(45,48)(46,47);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,37,38,45)(34,39,40,46)(35,43,41,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=f,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, 5036356);