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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3845 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(2,6)(3,4)(5,13)(15,16)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8937197);