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 | |
4278 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34)(35,45)(36,37)(38,42)(39,47)(40,44)(41,48)(43,46);(4,11)(8,14)(10,15)(12,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,31)(30,32)(33,35)(34,41)(36,46)(37,47)(38,39)(40,48)(42,43)(44,45);(2,7)(3,9)(8,14)(10,15)(18,23)(19,24)(25,29)(27,32)(33,36,40,38)(34,42,44,37)(35,46,48,39)(41,43,45,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30)(33,37,40,42)(34,36,44,38)(35,47,48,43)(39,41,46,45);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,38,40,36)(34,42,44,37)(35,46,48,39)(41,47,45,43);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,32)(26,30)(33,39,40,46)(34,43,44,47)(35,38,48,36)(37,41,42,45)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624793);