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 | |
1352 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)(10,13)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(11,15)(14,16)(19,26)(21,29)(23,25)(24,28)(33,35,38,40)(34,39,42,36)(37,43,47,46)(41,48,45,44);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35)(34,36)(37,46)(38,40)(39,42)(41,48)(43,47)(44,45);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgh,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, 5044585);