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 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,8,11)(5,13,15,9)(10,14,16,12)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,46)(44,47)(45,48);(2,7)(4,11)(9,13)(12,14)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,48,44,46);(2,7)(3,8)(9,13)(10,16)(18,24)(20,25)(23,30)(26,31)(34,36)(35,42)(41,44)(43,45);(1,3,6,8)(2,4,7,11)(5,10,15,16)(9,12,13,14)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,36,39,34)(35,38,42,40)(37,44,47,41)(43,48,45,46);(1,4)(2,8)(3,7)(5,14)(6,11)(9,10)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,47)(34,41,36,44)(35,43,42,45)(38,48,40,46);(1,5,6,15)(2,9,7,13)(3,10,8,16)(4,12,11,14)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,38)(34,35)(36,42)(37,46)(39,40)(41,45)(43,44)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=i,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, 9624957);