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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(6,8)(9,10,13,15)(11,12,16,14)(17,18,23,28)(19,24,26,21)(20,25,34,40)(22,27,36,42)(29,37,31,32)(30,38,41,35)(33,39,46,48)(43,47,44,45);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(19,26)(21,24)(22,36)(27,42)(29,31)(32,37)(33,46)(39,48);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,16)(14,15)(17,19,23,26)(18,24,28,21)(20,31,34,29)(22,30,36,41)(25,32,40,37)(27,38,42,35)(33,44,46,43)(39,45,48,47);(9,12)(10,11)(13,14)(15,16)(17,20)(18,25)(19,29)(21,32)(22,33)(23,34)(24,37)(26,31)(27,39)(28,40)(30,43)(35,45)(36,46)(38,47)(41,44)(42,48);(1,3)(2,6)(4,8)(5,7)(17,21,23,24)(18,26,28,19)(20,32,34,37)(22,35,36,38)(25,31,40,29)(27,41,42,30)(33,45,46,47)(39,44,48,43);(1,4,5,2)(3,8,7,6)(17,22)(18,27)(19,30)(20,33)(21,35)(23,36)(24,38)(25,39)(26,41)(28,42)(29,43)(31,44)(32,45)(34,46)(37,47)(40,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,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, 8664552);