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 | |
4112 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,25)(20,29)(21,30)(22,24)(23,28)(26,27)(31,32)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,31)(30,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,6)(7,14)(8,11)(15,16)(18,24)(20,30)(23,25)(26,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20)(18,21)(19,26)(22,30)(23,27)(24,29)(25,31)(28,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19)(18,25)(20,26)(21,31)(22,28)(23,24)(27,29)(30,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,21)(18,20)(19,27)(22,29)(23,26)(24,30)(25,32)(28,31)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9743086);