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 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,12)(13,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,15)(14,16)(18,22)(20,28)(23,25)(26,32)(36,43)(39,41)(44,48)(46,47);(3,4)(6,8)(10,14)(11,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4,5,3)(2,6,7,8)(9,10)(11,16)(12,15)(13,14)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,2)(3,8)(4,6)(5,7)(9,12,13,16)(10,15,14,11)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=fghi,d^2=ghi,e^2=fh,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, 6038563);