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 | |
4120 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)(8,13)(9,14)(10,16)(12,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34,39,40)(35,38,36,43)(37,41,45,48)(42,47,44,46);(2,6)(3,9)(8,11)(10,12)(20,27)(24,30)(26,31)(28,32)(33,35)(34,38)(36,39)(37,44)(40,43)(41,46)(42,45)(47,48);(2,6)(3,9)(8,11)(10,12)(18,23)(19,25)(24,30)(26,31)(34,40)(35,36)(37,45)(46,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,35)(34,38,40,43)(37,44,45,42)(41,47,48,46);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37)(34,41)(35,42)(36,44)(38,47)(39,45)(40,48)(43,46);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,38)(34,35)(36,40)(37,46)(39,43)(41,44)(42,48)(45,47)Polycyclic presentation:
2^9|a^2=gh,d^2=i,e^2=g,f^2=g,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, 9944022);