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 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,12,7)(4,13,14,9)(5,10,15,16)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(34,38)(39,45)(40,43)(46,47);(1,3,6,12)(2,7,8,11)(4,10,14,16)(5,9,15,13)(17,19,23,26)(18,24,22,21)(20,27,29,32)(25,31,30,28)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(2,8)(4,14)(7,11)(10,16)(19,26)(21,24)(27,32)(28,31)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,3)(2,7)(4,10)(5,13)(6,12)(8,11)(9,15)(14,16)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,4,6,14)(2,9,8,13)(3,10,12,16)(5,7,15,11)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,45)(40,46)(43,47)(44,48);(1,5,6,15)(2,10,8,16)(3,9,12,13)(4,7,14,11)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,36)(34,40)(35,41)(37,44)(38,43)(39,46)(42,48)(45,47)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8459289);