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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,9)(6,8)(7,11)(10,12)(13,15)(14,16)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29)(34,38)(36,44)(39,41)(42,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(18,24)(21,25)(26,32)(29,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,8)(4,7)(9,16)(12,15)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,30,32,29)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,6,7)(2,3,8,11)(5,12,14,15)(9,10,16,13)(17,20,23,19)(18,21,24,25)(22,28,31,27)(26,29,32,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,6,14)(2,9,8,16)(3,10,11,13)(4,12,7,15)(17,21)(18,19)(20,24)(22,30)(23,25)(26,28)(27,32)(29,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3)(2,7)(4,8)(5,13)(6,11)(9,12)(10,14)(15,16)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9831220);