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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,40,44)(35,45,38,41)(36,42,37,46)(39,43,48,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(19,25)(22,29)(26,31)(30,32)(33,35,40,38)(34,41,44,45)(36,43,37,47)(39,46,48,42);(2,6)(7,9)(8,12)(14,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,40,37)(34,42,44,46)(35,43,38,47)(39,45,48,41);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,36)(34,42,44,46)(35,43,38,47)(39,41,48,45);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,38,40,35)(34,41,44,45)(36,47,37,43)(39,46,48,42);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,39)(34,43)(35,46)(36,45)(37,41)(38,42)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=g,f^2=h,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, 10154711);