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 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(19,25)(20,27)(22,28)(23,30)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,2,6,4)(3,7,5,8)(9,10,12,14)(11,15,16,13)(19,25)(22,28)(26,31)(29,32)(34,35)(37,45)(39,41)(43,48);(1,3,6,5)(2,7,4,8)(9,10,12,14)(11,13,16,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,4,6,2)(3,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,6,3)(2,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=ghi,d^2=fgh,e^2=fgh,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, 6159869);