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 | |
4169 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)(5,10)(6,7)(9,13)(11,16)(12,15)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(34,36)(35,38)(41,44)(43,45);(2,7)(5,14)(8,9)(12,16)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3,6,13)(2,8,7,9)(4,11,10,15)(5,12,14,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,9)(4,15)(5,12)(6,13)(7,8)(10,11)(14,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,6,10)(2,5,7,14)(3,11,13,15)(8,12,9,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,10)(3,12)(4,7)(6,14)(8,15)(9,11)(13,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,e^2=gh,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=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297654);