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 |
| prime radical | #52839 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,10,5,8)(6,12,14,9)(11,15,13,16)(17,18,22,25)(19,26,24,21)(20,29,31,23)(27,30,28,32)(35,41)(37,45)(38,42)(40,44);(1,2,7,4)(3,8,5,10)(6,9,14,12)(11,15,13,16)(17,19)(18,21)(20,27)(22,24)(23,30)(25,26)(28,31)(29,32)(33,34,39,36)(35,42,41,38)(37,44,45,40)(43,46,47,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,21,22,26)(18,24,25,19)(20,30,31,32)(23,28,29,27)(33,36,39,34)(35,42,41,38)(37,40,45,44)(43,46,47,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,20)(18,23)(19,28)(21,32)(22,31)(24,27)(25,29)(26,30)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297688);