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 | |
4083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,4,11,8)(6,9,14,13)(10,12,16,15)(35,39)(36,42)(43,45)(44,47);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(6,14)(9,13)(10,16)(12,15)(18,23)(21,25)(24,31)(27,32)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,6,7,14)(2,9,5,13)(3,10,11,16)(4,12,8,15)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8278721);