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 | |
4184 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34)(35,40)(36,42)(37,41)(38,39)(43,47)(44,48)(45,46);(2,7)(3,11)(4,14)(13,15)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31)(34,39)(40,42)(41,46)(47,48);(2,7)(4,14)(8,10)(12,16)(19,27)(21,24)(28,32)(29,30)(33,35,38,36)(34,40,39,42)(37,43,45,44)(41,47,46,48);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,36,38,35)(34,40,39,42)(37,43,45,44)(41,48,46,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,37,38,45)(34,41,39,46)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8494659);