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 | |
4285 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,38,41)(35,36,39,42)(37,40,46,48)(43,44,47,45);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(2,3)(7,9)(8,10)(14,15)(18,24)(20,25)(23,30)(26,31)(35,39)(36,42)(43,47)(44,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,45)(44,47)(46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,36,38,42)(34,39,41,35)(37,45,46,44)(40,43,48,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,48)(42,45)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=h,e^2=i,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, 9963098);