Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,7)(6,8)(10,12)(14,15)(17,18)(19,25)(20,26)(21,27)(22,28)(23,24)(29,32)(30,41)(31,35)(33,42)(34,39)(36,37)(38,40)(43,45)(44,47)(46,48);(1,2,6,8)(3,5,7,4)(9,10,13,12)(11,15,16,14)(18,24)(21,25)(26,39)(28,40)(32,35)(36,41)(42,48)(45,47);(1,3,6,7)(2,4,8,5)(9,11,13,16)(10,14,12,15)(17,19,23,27)(18,25,24,21)(20,31,34,29)(22,37,38,30)(26,35,39,32)(28,36,40,41)(33,43,46,44)(42,45,48,47);(1,4,6,5)(2,3,8,7)(9,12,13,10)(11,15,16,14)(17,20,23,34)(18,26,24,39)(19,29,27,31)(21,35,25,32)(22,33,38,46)(28,42,40,48)(30,43,37,44)(36,47,41,45);(1,2)(3,4)(5,7)(6,8)(17,21,23,25)(18,27,24,19)(20,32,34,35)(22,36,38,41)(26,29,39,31)(28,37,40,30)(33,45,46,47)(42,43,48,44);(1,5,6,4)(2,7,8,3)(17,22)(18,28)(19,30)(20,33)(21,36)(23,38)(24,40)(25,41)(26,42)(27,37)(29,43)(31,44)(32,45)(34,46)(35,47)(39,48)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839338);