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 | |
4247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,5)(6,8)(7,13)(9,15)(11,12)(14,16)(17,18,23,27)(19,24,28,32)(20,21,25,30)(22,26,31,29)(35,39)(36,42)(43,45)(44,47);(1,3,6,13)(2,7,8,10)(4,14,15,11)(5,12,9,16)(17,19)(18,24)(20,22)(21,26)(23,28)(25,31)(27,32)(29,30)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(2,8)(5,9)(7,10)(12,16)(19,28)(22,31)(24,32)(26,29)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,3)(2,7)(4,14)(5,12)(6,13)(8,10)(9,16)(11,15)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,31)(29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,9)(3,11)(5,8)(6,15)(7,12)(10,16)(13,14)(17,21)(18,25)(19,29)(20,27)(22,24)(23,30)(26,28)(31,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5,6,9)(2,4,8,15)(3,12,13,16)(7,14,10,11)(17,22)(18,26)(19,20)(21,24)(23,31)(25,28)(27,29)(30,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8256116);