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 | |
4161 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)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18,24,22)(19,21,25,20)(23,30,31,26)(27,28,32,29)(33,34,39,43)(35,40,45,37)(36,41,38,42)(44,48,46,47);(17,19)(18,20)(21,22)(23,27)(24,25)(26,28)(29,30)(31,32)(33,35,39,45)(34,40,43,37)(36,46,38,44)(41,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,24,21)(18,25,22,19)(23,28,31,29)(26,32,30,27)(33,36,39,38)(34,41,43,42)(35,44,45,46)(37,47,40,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21,24,20)(18,25,22,19)(23,29,31,28)(26,32,30,27)(33,34)(35,37)(36,41)(38,42)(39,43)(40,45)(44,47)(46,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,22,24,18)(19,21,25,20)(23,26,31,30)(27,28,32,29)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,46)(43,45);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,23)(18,26)(19,27)(20,28)(21,29)(22,30)(24,31)(25,32)(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=i,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, 10441516);