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 | |
4229 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)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,42)(36,38)(37,40)(39,41)(43,44)(45,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,39,37)(34,40,41,42)(36,45,46,43)(38,44,48,47);(3,10)(6,7)(11,16)(14,15)(20,28)(25,30)(27,29)(31,32)(34,41)(36,46)(40,42)(43,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36,39,46)(34,38,41,48)(35,43,37,45)(40,47,42,44);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37,39,35)(34,42,41,40)(36,45,46,43)(38,44,48,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38)(34,36)(35,44)(37,47)(39,48)(40,45)(41,46)(42,43)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635951);