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 | |
4319 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)(17,18,23,27)(19,26,30,24)(20,22,21,25)(28,29,31,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,48);(3,10)(6,9)(11,12)(15,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27)(33,35,39,40)(34,37,42,36)(38,43,47,48)(41,45,46,44);(1,3)(2,6)(4,11)(5,10)(7,15)(8,9)(12,14)(13,16)(17,20,23,21)(18,25,27,22)(19,28,30,31)(24,29,26,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,3)(2,6)(4,12)(5,10)(7,16)(8,9)(11,14)(13,15)(17,21,23,20)(18,22,27,25)(19,28,30,31)(24,29,26,32)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,48,46,43);(1,4,5,14)(2,7,8,13)(3,11,10,12)(6,15,9,16)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,44)(45,48);(1,2)(3,6)(4,13)(5,8)(7,14)(9,10)(11,16)(12,15)(17,22)(18,20)(19,29)(21,27)(23,25)(24,31)(26,28)(30,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=gh,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, 10371741);