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 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,6)(3,5,9,4)(7,14,15,10)(11,12,16,13)(17,18)(19,23)(20,25)(21,31)(22,27)(24,30)(26,28)(29,32)(33,34,39,43)(35,40,45,37)(36,41,38,42)(44,48,46,47);(1,3)(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,19,22,29)(18,23,27,32)(20,28,30,21)(24,31,25,26)(33,35,39,45)(34,40,43,37)(36,46,38,44)(41,47,42,48);(1,4,8,5)(2,9,6,3)(7,12,15,13)(10,16,14,11)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32)(33,36,39,38)(34,41,43,42)(35,44,45,46)(37,47,40,48);(1,5,8,4)(2,9,6,3)(7,13,15,12)(10,16,14,11)(17,20)(18,25)(19,28)(21,29)(22,30)(23,26)(24,27)(31,32)(33,34)(35,37)(36,41)(38,42)(39,43)(40,45)(44,47)(46,48);(1,6,8,2)(3,5,9,4)(7,10,15,14)(11,12,16,13)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,46)(43,45);(1,7)(2,10)(3,11)(4,12)(5,13)(6,14)(8,15)(9,16)(17,21,22,28)(18,26,27,31)(19,20,29,30)(23,24,32,25)(33,38,39,36)(34,42,43,41)(35,44,45,46)(37,47,40,48)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441538);