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 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,4)(5,13,16,9)(11,12,14,15)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3)(2,7)(4,8)(5,14)(6,10)(9,12)(11,16)(13,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(34,35)(39,41)(40,42)(46,47);(2,8)(3,10)(9,13)(11,14)(18,23)(21,29)(24,25)(27,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,6,7)(2,3,8,10)(5,15,16,12)(9,14,13,11)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,6,16)(2,9,8,13)(3,11,10,14)(4,12,7,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,3)(2,7)(4,8)(5,11)(6,10)(9,15)(12,13)(14,16)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300657);