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 | |
4053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,20)(19,24,27,28)(21,25,31,29)(22,30,32,26)(34,38)(36,44)(39,41)(42,48);(1,2,6,7)(3,5,9,14)(4,8,11,15)(10,13,12,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,7)(4,11)(5,14)(10,12)(18,20)(19,27)(21,31)(22,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3,6,9)(2,5,7,14)(4,12,11,10)(8,16,15,13)(17,20,23,18)(19,28,27,24)(21,25,31,29)(22,30,32,26)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21)(18,25)(19,26)(20,29)(22,24)(23,31)(27,30)(28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5)(2,9)(3,7)(4,13)(6,14)(8,12)(10,15)(11,16)(17,22,23,32)(18,26,20,30)(19,29,27,25)(21,24,31,28)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9416099);