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 | |
3945 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,29)(28,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,15,14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(4,12)(8,14)(11,13)(15,16)(17,18,22,26)(19,23,30,27)(20,24,29,31)(21,25,28,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,4,6,12)(2,8,9,14)(3,11,5,13)(7,15,10,16)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,15,14,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,2)(3,7)(4,14)(5,10)(6,9)(8,12)(11,16)(13,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646568);