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 | |
4168 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)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(34,39)(36,40)(41,45)(44,47);(2,6)(4,13)(7,9)(11,16)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(33,34,38,39)(35,40,43,36)(37,45,46,41)(42,44,48,47);(2,6)(3,10)(8,12)(11,16)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669265);