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 | |
4043 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)(3,5)(4,12)(6,8)(7,11)(9,13)(10,15)(14,16)(17,18,22,21)(19,23,27,26)(20,28,30,24)(25,32,29,31);(1,3,6,11)(2,7,8,5)(4,10,13,16)(9,15,12,14)(18,21)(23,26)(24,28)(31,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,8)(4,13)(5,7)(10,16)(17,19,22,27)(18,23,21,26)(20,29,30,25)(24,32,28,31)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,4)(2,9)(3,10)(5,14)(6,13)(7,15)(8,12)(11,16)(17,20,22,30)(18,24,21,28)(19,25,27,29)(23,31,26,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,6,7)(2,3,8,11)(4,14,13,15)(9,10,12,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,14)(13,16)(17,21,22,18)(19,26,27,23)(20,28,30,24)(25,32,29,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8690432);