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 | |
3979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,11)(10,15)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,48,47,46);(1,2,5,6)(3,7,9,13)(4,11,12,8)(10,16,15,14)(17,19,22,20)(18,23,24,25)(21,28,30,27)(26,32,29,31)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(3,9)(7,13)(10,15)(14,16)(18,24)(23,25)(26,29)(31,32)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,46,41,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,30)(18,26,24,29)(19,27,20,28)(23,31,25,32)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662423);