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 | |
4102 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)(3,7)(4,9)(5,13)(6,8)(10,15)(11,12)(14,16)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,6,9)(2,7,8,4)(5,11,15,16)(10,14,13,12)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,8)(3,9)(10,13)(11,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,16)(13,15)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,4,6,7)(2,9,8,3)(5,14,15,12)(10,11,13,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5)(2,10)(3,11)(4,12)(6,15)(7,14)(8,13)(9,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9757991);