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 | |
3971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18,23,25)(19,24,27,32)(20,22,30,31)(21,29,28,26)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(1,3,6,4)(2,7,8,9)(5,12,14,11)(10,16,13,15)(17,19,23,27)(18,24,25,32)(20,28,30,21)(22,26,31,29)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(2,8)(7,9)(10,13)(15,16)(18,25)(19,27)(20,30)(26,29)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,20)(18,22)(19,28)(21,27)(23,30)(24,26)(25,31)(29,32)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,21,23,28)(18,26,25,29)(19,20,27,30)(22,24,31,32)(33,37,39,46)(34,41,36,44)(35,42,43,48)(38,45,40,47);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,22)(18,20)(19,29)(21,24)(23,31)(25,30)(26,27)(28,32)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662633);