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 | |
4039 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)(3,5)(4,12)(6,7)(8,11)(9,15)(10,16)(13,14)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(35,41)(37,45)(38,42)(40,44);(2,7)(5,8)(9,12)(14,16)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34,39,36)(35,42,41,38)(37,44,45,40)(43,46,47,48);(1,3,6,11)(2,5,7,8)(4,10,15,13)(9,14,12,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,3)(2,8)(4,13)(5,7)(6,11)(9,14)(10,15)(12,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,34)(35,42,41,38)(37,40,45,44)(43,46,47,48);(1,4,6,15)(2,9,7,12)(3,10,11,13)(5,14,8,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,5,6,8)(2,3,7,11)(4,14,15,16)(9,10,12,13)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,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, 9297686);