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 | |
1367 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,32)(28,31)(34,35)(36,43)(37,45)(46,47);(1,2)(3,8)(4,7)(5,6)(11,16)(13,15)(18,23)(20,27)(21,30)(26,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,6)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,32)(30,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,3,5,7)(2,4,6,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,4)(2,7)(3,6)(5,8)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fghi,d^2=fhi,e^2=gh,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, 6038172);