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 | |
4089 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)(3,4)(5,13)(6,7)(8,15)(9,10)(11,14)(12,16)(18,22)(20,28)(23,25)(26,29)(33,34)(35,42)(36,41)(37,45)(38,43)(39,44)(40,46)(47,48);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(33,35)(34,40)(36,43)(37,47)(38,45)(39,46)(41,48)(42,44);(2,7)(3,9)(8,13)(11,16)(17,18,21,22)(19,23,27,25)(20,24,28,30)(26,31,29,32)(33,36,39,37)(34,41,44,45)(35,43,46,47)(38,40,48,42);(1,2)(3,10)(4,9)(5,8)(6,7)(11,14)(12,16)(13,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,37,39,36)(34,41,44,45)(35,43,46,47)(38,42,48,40);(1,4,6,10)(2,3,7,9)(5,14,15,12)(8,16,13,11)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,35)(34,42)(36,43)(37,47)(38,41)(39,46)(40,44)(45,48);(1,5)(2,8)(3,11)(4,12)(6,15)(7,13)(9,16)(10,14)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,38)(34,43)(35,45)(36,40)(37,42)(39,48)(41,46)(44,47)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154340);