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 | |
4272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,9,12,13)(11,16,14,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34,39,43)(35,44,37,40)(36,41,42,38)(45,48,47,46);(1,3,7,5)(2,8,6,10)(4,11,12,14)(9,15,13,16)(17,19,22,26)(18,23,25,20)(21,27,30,32)(24,31,29,28)(33,34)(35,40)(36,41)(37,44)(38,42)(39,43)(45,46)(47,48);(4,12)(9,13)(11,14)(15,16)(19,26)(20,23)(27,32)(28,31)(33,35,39,37)(34,40,43,44)(36,47,42,45)(38,46,41,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,36,39,42)(34,41,43,38)(35,45,37,47)(40,46,44,48);(1,5,7,3)(2,8,6,10)(4,11,12,14)(9,16,13,15)(17,18)(19,23)(20,26)(21,29)(22,25)(24,30)(27,28)(31,32)(33,37,39,35)(34,40,43,44)(36,45,42,47)(38,46,41,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,41)(34,42,43,36)(35,46,37,48)(40,45,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9082224);