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 | |
4027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,7,14,13)(11,16,12,15)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,39,42)(35,36,40,38)(37,41,45,48)(43,44,47,46);(3,10)(6,9)(11,12)(15,16)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,35,39,40)(34,38,42,36)(37,43,45,47)(41,46,48,44);(1,3)(2,6)(4,11)(5,10)(7,15)(8,9)(12,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,36,39,38)(34,40,42,35)(37,44,45,46)(41,47,48,43);(1,3)(2,6)(4,12)(5,10)(7,16)(8,9)(11,14)(13,15)(17,18)(19,21)(20,24)(22,25)(23,28)(26,29)(27,31)(30,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,4,5,14)(2,7,8,13)(3,11,10,12)(6,15,9,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,2)(3,6)(4,13)(5,8)(7,14)(9,10)(11,16)(12,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,38,39,36)(34,35,42,40)(37,44,45,46)(41,47,48,43)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10371752);