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 | |
4077 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)(3,10,8,4)(5,9,14,15)(11,16,13,12)(18,19)(23,25)(24,26)(30,31)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(2,7)(3,8)(9,15)(11,13)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(34,40)(35,36)(42,45)(43,44);(2,7)(4,10)(9,15)(12,16)(20,27)(23,25)(28,32)(30,31)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3,6,8)(2,4,7,10)(5,13,14,11)(9,16,15,12)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4)(2,8)(3,7)(5,12)(6,10)(9,13)(11,15)(14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,14)(2,9,7,15)(3,11,8,13)(4,12,10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442127);