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 | |
4234 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(18,23)(19,24)(25,30)(26,31)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,43,44,38);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35,39,47)(34,40,45,48)(36,44,46,37)(38,41,43,42);(2,6)(3,10)(4,13)(14,15)(19,24)(20,27)(21,29)(25,30)(33,36,39,46)(34,41,45,42)(35,37,47,44)(38,48,43,40);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,39,44)(34,43,45,38)(35,46,47,36)(40,41,48,42);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,43)(34,44,45,37)(35,41,47,42)(36,40,46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8922466);