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 | |
3937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,4,11,7)(5,13,15,9)(10,16,14,12)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34,40,37)(35,39,42,44)(36,41,47,38)(43,45,46,48);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,42)(41,48)(43,47);(2,8)(4,7)(9,13)(12,16)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,36,40,47)(34,41,37,38)(35,43,42,46)(39,45,44,48);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,34)(35,44,42,39)(36,38,47,41)(43,48,46,45);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,38)(34,36)(35,45)(37,47)(39,46)(40,41)(42,48)(43,44);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,44)(34,42,37,35)(36,45,47,48)(38,43,41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=ghi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619487);