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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34,38,41)(35,39,36,43)(37,45,46,40)(42,48,44,47);(2,5)(6,8)(11,15)(13,16)(18,22)(19,26)(20,29)(30,31)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,2,4,5)(3,6,7,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=gh,d^2=fgi,e^2=f,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, 5971241);