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 |
| prime radical | #52357 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3949 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,22,26)(19,25,30,23)(20,21,31,24)(27,32,28,29)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,29)(22,30)(24,32)(25,26)(28,31)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(2,3)(7,9)(8,10)(15,16)(17,20,22,31)(18,24,26,21)(19,27,30,28)(23,32,25,29)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,21)(19,28)(22,31)(23,32)(24,26)(25,29)(27,30)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,20)(19,29)(22,24)(23,27)(25,28)(26,31)(30,32)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,19)(18,25)(20,27)(21,32)(22,30)(23,26)(24,29)(28,31)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=gh,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, 8646456);