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 | |
4087 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,7)(3,9)(10,15)(11,13)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(34,35)(39,41)(40,42)(46,47);(2,7)(4,8)(10,15)(12,16)(18,25)(19,21)(26,32)(27,29)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441868);