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 | |
4060 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)(3,9)(4,13)(5,7)(6,10)(8,11)(12,15)(14,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,27,28,31)(34,39)(35,41)(37,45)(44,47);(1,3,8,12)(2,9,11,15)(4,14,5,10)(6,13,16,7)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,34,38,39)(35,40,41,36)(37,42,45,46)(43,47,48,44);(1,4,8,5)(2,7,11,13)(3,10,12,14)(6,9,16,15)(17,20,22,30)(18,24,25,21)(19,26,29,28)(23,27,32,31)(33,34,38,39)(35,40,41,36)(37,46,45,42)(43,44,48,47);(1,5,8,4)(2,7,11,13)(3,10,12,14)(6,15,16,9)(17,21)(18,20)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,6)(2,10)(3,7)(4,15)(5,9)(8,16)(11,14)(12,13)(17,20)(18,24)(19,28)(21,25)(22,30)(23,31)(26,29)(27,32)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,7)(2,4)(3,6)(5,11)(8,13)(9,14)(10,15)(12,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,37,38,45)(34,42,39,46)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851191);