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 | |
4010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,8,14,16)(4,5,10,9)(6,15,13,11)(17,18,23,27)(19,24,21,31)(20,25,26,22)(28,30,29,32)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(1,3,7,14)(2,8,12,16)(4,13,10,6)(5,11,9,15)(17,19)(18,24)(20,29)(21,23)(22,30)(25,32)(26,28)(27,31)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4)(2,9)(3,13)(5,12)(6,14)(7,10)(8,15)(11,16)(17,20)(18,22)(19,28)(21,29)(23,26)(24,32)(25,27)(30,31)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,4)(2,5)(3,13)(6,14)(7,10)(8,11)(9,12)(15,16)(17,21,23,19)(18,24,27,31)(20,29,26,28)(22,32,25,30)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5)(2,10)(3,11)(4,12)(6,8)(7,9)(13,16)(14,15)(17,20)(18,25)(19,29)(21,28)(22,27)(23,26)(24,32)(30,31)(33,37,39,35)(34,40,36,41)(38,43,46,45)(42,48,44,47);(1,6,7,13)(2,11,12,15)(3,4,14,10)(5,8,9,16)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,28,31,29)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10395520);