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 | |
3898 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34,39,36)(35,42,41,38)(37,44,46,40)(43,45,47,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,34,39,36)(35,42,41,38)(37,44,46,40)(43,45,47,48);(2,7)(3,10)(8,15)(11,12)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,36,39,34)(35,42,41,38)(37,40,46,44)(43,45,47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=gh,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=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297629);