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 | |
4020 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,27)(20,30)(21,31)(22,26)(23,28)(24,29)(25,32)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(19,28)(21,32)(23,27)(25,31)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(3,12)(6,15)(8,16)(10,13)(17,19,22,28)(18,23,26,27)(20,21,29,32)(24,31,30,25)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20)(18,24)(19,21)(22,29)(23,31)(25,27)(26,30)(28,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,21)(18,25)(19,29)(20,28)(22,32)(23,24)(26,31)(27,30)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,30)(31,32)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313838);