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 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,7,14,13)(11,16,12,15)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,40,38)(35,37,41,36)(39,46,47,42)(43,44,48,45);(3,10)(6,9)(11,12)(15,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,36)(37,38)(39,43)(40,41)(42,44)(45,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,9)(12,14)(13,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36,40,37)(34,41,38,35)(39,44,47,45)(42,48,46,43);(1,3)(2,6)(4,12)(5,10)(7,16)(8,9)(11,14)(13,15)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,37,40,36)(34,41,38,35)(39,45,47,44)(42,48,46,43);(1,4,5,14)(2,7,8,13)(3,11,10,12)(6,15,9,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,38,40,34)(35,37,41,36)(39,42,47,46)(43,44,48,45);(1,2)(3,6)(4,13)(5,8)(7,14)(9,10)(11,16)(12,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,39,40,47)(34,42,38,46)(35,43,41,48)(36,44,37,45)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=h,f^2=g,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, 10371760);