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 | |
4057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,6)(4,10,14,16)(11,13,12,15)(17,18)(19,27)(20,29)(21,28)(22,26)(23,24)(25,31)(30,32)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(1,3)(2,8)(4,12)(5,7)(6,9)(10,15)(11,14)(13,16)(17,19,23,28)(18,21,24,27)(20,30,31,22)(25,32,29,26)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,9)(3,5)(10,16)(11,12)(18,24)(21,27)(25,29)(26,32)(35,40)(37,45)(43,46)(44,48);(1,4)(2,10)(3,11)(5,12)(6,15)(7,14)(8,13)(9,16)(17,20,23,31)(18,25,24,29)(19,22,28,30)(21,26,27,32)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,5,7,3)(2,8,9,6)(4,12,14,11)(10,13,16,15)(17,21)(18,19)(20,26)(22,25)(23,27)(24,28)(29,30)(31,32)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,6,7,8)(2,3,9,5)(4,13,14,15)(10,12,16,11)(17,22)(18,26)(19,20)(21,25)(23,30)(24,32)(27,29)(28,31)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529202);