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 | |
4203 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,38,41)(35,39,42,36)(37,40,46,48)(43,47,45,44);(2,6)(7,9)(8,12)(15,16)(17,19,22,27)(18,20,23,25)(21,30,31,26)(24,32,29,28)(33,35,38,42)(34,39,41,36)(37,45,46,43)(40,44,48,47);(3,10)(7,9)(11,13)(15,16)(18,23)(20,25)(24,29)(28,32)(35,42)(36,39)(43,45)(44,47);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,20,22,25)(18,19,23,27)(21,28,31,32)(24,26,29,30)(33,36)(34,35)(37,44)(38,39)(40,43)(41,42)(45,48)(46,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,48)(42,45)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8735827);