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 | |
4183 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34)(35,43)(36,47)(37,41)(38,42)(39,46)(40,45)(44,48);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(19,26)(22,25)(27,28)(31,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(3,11)(4,12)(7,16)(8,13)(17,19,21,26)(18,22,24,25)(20,27,30,28)(23,31,29,32)(33,34,38,42)(35,39,46,43)(36,40,45,47)(37,41,44,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,34)(35,43)(36,40)(37,48)(38,42)(39,46)(41,44)(45,47);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,36)(34,40)(35,44)(37,46)(38,45)(39,48)(41,43)(42,47);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,29)(25,32)(26,28)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641353);