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 | |
4067 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)(4,9)(6,8)(7,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32)(33,34,39,38)(35,42,45,40)(36,37,41,47)(43,48,46,44);(1,3,6,10)(2,7,8,5)(4,11,12,16)(9,15,13,14)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,46)(47,48);(2,8)(4,12)(5,7)(11,16)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,36,39,41)(34,37,38,47)(35,43,45,46)(40,44,42,48);(1,2)(3,7)(4,13)(5,10)(6,8)(9,12)(11,14)(15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,37,39,47)(34,41,38,36)(35,44,45,48)(40,46,42,43);(1,4,6,12)(2,9,8,13)(3,11,10,16)(5,14,7,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,38,39,34)(35,40,45,42)(36,37,41,47)(43,48,46,44);(1,5,6,7)(2,10,8,3)(4,14,12,15)(9,16,13,11)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(33,35)(34,42)(36,46)(37,44)(38,40)(39,45)(41,43)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957577);