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 | |
1379 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,16)(14,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(35,41)(38,42)(43,47)(45,48);(1,3)(2,5)(4,8)(6,7)(10,15)(12,13)(20,27)(23,25)(28,32)(30,31)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(1,2,4,7)(3,6,8,5)(9,11)(10,12)(13,15)(14,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,12,14,13)(10,16,15,11)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,35)(34,38)(36,43)(37,41)(39,45)(40,42)(44,47)(46,48);(1,3)(2,6)(4,8)(5,7)(9,13,14,12)(10,11,15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,44)(34,39,40,46)(35,43,41,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=hi,d^2=ghi,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, 6038754);