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 | |
4142 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)(18,22)(20,28)(23,25)(26,32)(33,34)(35,46)(36,42)(37,47)(38,43)(39,44)(40,45)(41,48);(3,9)(7,13)(10,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,40,48)(34,41,45,46)(36,38,37,44)(39,47,43,42);(2,6)(7,13)(8,11)(14,16)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,40,37)(34,42,45,47)(35,38,48,44)(39,46,43,41);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,40,36)(34,42,45,47)(35,38,48,44)(39,41,43,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,38)(34,43)(35,37)(36,48)(39,45)(40,44)(41,42)(46,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,39,40,43)(34,44,45,38)(35,47,48,42)(36,46,37,41)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=gi,e^2=h,f^2=g,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, 10153992);