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 | |
4303 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)(3,8)(4,10)(5,14)(6,13)(7,9)(11,15)(12,16)(18,22)(19,25)(24,30)(26,27)(36,44)(39,46)(42,48)(45,47);(1,3)(2,8)(4,7)(5,13)(6,14)(9,10)(11,16)(12,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(2,9)(3,4)(11,14)(12,13)(19,25)(23,29)(26,27)(31,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,4,7,3)(2,10,9,8)(5,13,15,12)(6,11,16,14)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,5)(2,11)(3,13)(4,12)(6,10)(7,15)(8,16)(9,14)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,6)(2,12)(3,14)(4,11)(5,10)(7,16)(8,15)(9,13)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064177);