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 | |
4032 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,23)(20,21)(22,25)(24,26)(27,28)(29,32)(30,31)(34,39)(35,41)(37,45)(44,46);(2,7)(4,12)(8,14)(10,13)(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,30)(31,32)(34,39)(36,42)(40,47)(44,46);(2,7)(3,11)(9,16)(10,13)(19,26)(23,24)(27,29)(28,32)(33,34,38,39)(35,42,41,36)(37,40,45,47)(43,46,48,44);(1,3)(2,8)(4,13)(5,9)(6,11)(7,14)(10,12)(15,16)(17,19)(18,24)(20,29)(21,28)(22,26)(23,25)(27,30)(31,32)(33,35,38,41)(34,36,39,42)(37,43,45,48)(40,44,47,46);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,20,22,30)(18,21,25,31)(19,27,26,29)(23,28,24,32)(33,36)(34,35)(37,46)(38,42)(39,41)(40,48)(43,47)(44,45);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,21)(18,20)(19,28)(22,31)(23,27)(24,29)(25,30)(26,32)(33,37,38,45)(34,40,39,47)(35,43,41,48)(36,44,42,46)Polycyclic presentation:
2^9|c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 9110824);