Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,8,7)(11,15)(13,16)(19,25)(20,27)(22,28)(23,30)(34,38)(35,41)(40,45)(43,47);(2,6)(4,7)(9,10,12,14)(11,16,15,13)(19,25)(22,28)(26,31)(29,32)(34,38)(39,42)(40,45)(46,48);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4,5,7)(2,3,6,8)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=ghi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926523);