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 | |
3937 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)(3,12)(4,6)(5,10)(7,9)(8,14)(11,15)(13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,7,14)(2,8,9,12)(4,13,11,10)(5,15,16,6)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,9)(4,11)(8,12)(10,13)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,11)(2,6,9,15)(3,13,14,10)(5,12,16,8)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,5,7,16)(2,10,9,13)(3,6,14,15)(4,8,11,12)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,6)(2,11)(3,5)(4,9)(7,15)(8,10)(12,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091093);