Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(34,35)(36,43)(37,45)(46,47);(3,8)(5,7)(10,13)(11,15)(18,19)(20,27)(24,26)(28,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35,38,34)(36,41,43,39)(37,40,45,42)(44,46,48,47);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,2)(3,5)(4,6)(7,8)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6081876);