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 | |
4220 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,22)(20,23)(21,26)(24,30)(25,27)(28,32)(29,31)(33,34)(35,43)(36,45)(37,42)(38,46)(39,41)(40,44)(47,48);(3,9)(7,13)(10,15)(14,16)(18,19)(20,27)(24,26)(28,29)(33,35,39,44)(34,40,41,43)(36,37,46,47)(38,42,45,48);(2,6)(7,13)(8,11)(14,16)(18,19)(23,25)(24,26)(31,32)(34,41)(35,44)(36,46)(42,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,18)(20,23,27,25)(21,26,30,24)(28,31,29,32)(33,36)(34,38)(35,37)(39,46)(40,42)(41,45)(43,48)(44,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,37)(34,42)(35,36)(38,40)(39,47)(41,48)(43,45)(44,46);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38)(34,36)(35,42)(37,40)(39,45)(41,46)(43,47)(44,48)Polycyclic presentation:
2^9|b^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154571);