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 | |
4076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,14)(4,5,12,13)(9,10,15,16)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27)(33,34,39,36)(35,43,46,40)(37,41,45,47)(38,42,44,48);(1,3,6,11)(2,7,8,14)(4,9,12,15)(5,10,13,16)(17,19,22,24)(18,20,23,28)(21,26,30,31)(25,27,29,32)(33,35,39,46)(34,40,36,43)(37,38,45,44)(41,48,47,42);(2,8)(4,12)(7,14)(9,15)(18,23)(19,24)(21,30)(27,32)(33,34,39,36)(35,40,46,43)(37,41,45,47)(38,48,44,42);(1,4)(2,5)(3,9)(6,12)(7,10)(8,13)(11,15)(14,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,36,39,34)(35,43,46,40)(37,41,45,47)(38,48,44,42);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,5)(2,4)(3,10)(6,13)(7,9)(8,12)(11,16)(14,15)(17,21)(18,25)(19,26)(20,27)(22,30)(23,29)(24,31)(28,32)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385550);