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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(21,29)(24,26)(27,32)(30,31)(33,34,39,37)(35,43,46,40)(36,47,45,41)(38,48,44,42);(2,7)(3,10)(9,13)(11,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(2,7)(3,10)(9,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,37)(35,40,46,43)(36,47,45,41)(38,42,44,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,34)(35,43,46,40)(36,41,45,47)(38,48,44,42);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=gi,f^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, 8074638);