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 | |
4023 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,24,20,27)(22,29,30,25)(26,32,28,31)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(3,9)(7,13)(10,15)(14,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,6)(7,13)(8,11)(14,16)(18,21)(19,20)(25,29)(26,28)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,24,21,27)(22,28,30,26)(25,31,29,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,24,20,27)(22,25,30,29)(26,32,28,31)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,30)(18,25,21,29)(19,26,20,28)(24,31,27,32)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947948);