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 | |
4184 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32)(35,41)(39,42)(43,47)(46,48);(2,7)(4,13)(5,8)(11,14)(18,23)(21,25)(24,26)(29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(2,7)(3,10)(9,12)(11,14)(19,27)(24,26)(28,30)(29,32)(34,38)(39,42)(40,44)(46,48);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,32)(28,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,21,22,25)(18,20,23,31)(19,29,27,32)(24,30,26,28)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|d^2=h,e^2=h,f^2=g,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, 8732277);