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 | #51749 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(6,8)(9,10,13,14)(11,12,15,16)(17,18)(19,24)(20,31)(21,23)(22,26)(25,34)(27,39)(28,30)(29,38)(32,37)(33,44)(35,36)(40,46)(41,48)(42,43)(45,47);(3,7)(6,8)(10,14)(11,15)(18,21)(22,36)(24,28)(25,31)(29,43)(33,46)(37,39)(41,45);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,15)(14,16)(17,19,23,30)(18,24,21,28)(20,32,34,27)(22,29,36,43)(25,39,31,37)(26,38,35,42)(33,45,46,41)(40,48,44,47);(9,12)(10,11)(13,16)(14,15)(17,20)(18,25)(19,27)(21,31)(22,33)(23,34)(24,37)(26,40)(28,39)(29,41)(30,32)(35,44)(36,46)(38,47)(42,48)(43,45);(1,3)(2,6)(4,8)(5,7)(17,21,23,18)(19,28,30,24)(20,31,34,25)(22,35,36,26)(27,39,32,37)(29,42,43,38)(33,44,46,40)(41,48,45,47);(1,4,5,2)(3,8,7,6)(17,22)(18,26)(19,29)(20,33)(21,35)(23,36)(24,38)(25,40)(27,41)(28,42)(30,43)(31,44)(32,45)(34,46)(37,47)(39,48)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664551);