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 | |
4167 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)(5,10)(6,8)(9,14)(11,15)(12,16)(17,18,23,22)(19,26,21,24)(20,28,29,25)(27,31,30,32)(33,34)(35,42)(36,45)(37,47)(38,40)(39,41)(43,48)(44,46);(1,3,6,11)(2,7,8,15)(4,12,14,10)(5,13,16,9)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32)(33,35,39,38)(34,40,41,42)(36,46,47,43)(37,44,45,48);(2,8)(3,11)(9,13)(10,12)(20,29)(25,28)(27,30)(31,32)(34,41)(36,47)(40,42)(43,46);(1,3)(2,7)(4,12)(5,9)(6,11)(8,15)(10,14)(13,16)(17,20)(18,25)(19,27)(21,30)(22,28)(23,29)(24,31)(26,32)(33,36)(34,37)(35,43)(38,46)(39,47)(40,48)(41,45)(42,44);(1,4,6,14)(2,9,8,13)(3,12,11,10)(5,15,16,7)(17,21,23,19)(18,24,22,26)(20,30,29,27)(25,31,28,32)(33,37)(34,36)(35,44)(38,48)(39,45)(40,46)(41,47)(42,43);(1,5)(2,10)(3,13)(4,7)(6,16)(8,12)(9,11)(14,15)(17,22,23,18)(19,26,21,24)(20,25,29,28)(27,31,30,32)(33,38,39,35)(34,40,41,42)(36,43,47,46)(37,44,45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619667);