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 | |
4030 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)(4,7)(5,12)(6,10)(8,14)(11,15)(13,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(34,40)(35,38)(36,42)(45,47);(1,3,6,12)(2,5,10,9)(4,11,14,16)(7,13,8,15)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34)(35,43)(36,37)(38,41)(39,40)(42,48)(44,45)(46,47);(1,4,6,14)(2,7,10,8)(3,11,12,16)(5,13,9,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,48,47);(1,4)(2,8)(3,11)(5,15)(6,14)(7,10)(9,13)(12,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,42)(34,37,40,48)(35,44,38,46)(41,47,43,45);(1,5,6,9)(2,3,10,12)(4,13,14,15)(7,11,8,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37)(34,42)(35,45)(36,40)(38,47)(39,48)(41,44)(43,46);(1,3)(2,9)(4,11)(5,10)(6,12)(7,15)(8,13)(14,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,35)(34,41,40,43)(36,46,42,44)(37,47,48,45)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,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, 9297656);