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 | |
1302 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)(12,13)(15,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34,37,40)(35,41,43,38)(36,39,46,45)(42,48,44,47);(11,15)(14,16)(19,25)(23,26)(27,31)(30,32)(33,35,37,43)(34,38,40,41)(36,44,46,42)(39,48,45,47);(10,13)(14,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,36,37,46)(34,39,40,45)(35,42,43,44)(38,47,41,48)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=h,d^2=h,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, 5044430);