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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(1,3,4,7)(2,5,6,8)(9,11,13,10)(12,15,16,14)(18,22)(19,25)(24,28)(27,31)(37,45)(39,46)(42,48)(44,47);(1,3)(2,5)(4,7)(6,8)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,2)(3,5)(4,6)(7,8)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,c^2=g,d^2=i,e^2=f,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, 5017854);