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 |
| prime radical | #34589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(10,14)(11,16)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,2,6,7)(3,8,9,4)(5,10,13,14)(11,15,16,12)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31)(34,38)(35,42)(40,44)(43,48);(5,13)(10,14)(11,16)(12,15)(18,23)(21,25)(24,31)(27,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8278751);