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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,15)(12,21)(13,18)(14,19)(16,20)(17,23)(22,24)(25,26,30,32)(27,31,35,36)(28,39,38,33)(29,40,37,34);(1,2,5,4)(3,7,8,6)(9,11,14,20)(10,15,19,16)(12,22,23,18)(13,21,24,17)(25,27)(26,31)(28,37)(29,38)(30,35)(32,36)(33,40)(34,39);(1,3,5,8)(2,6,4,7)(11,20)(13,24)(15,16)(18,22)(26,32)(27,35)(29,37)(33,39);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,26)(27,36)(28,39)(29,34)(30,32)(31,35)(33,38)(37,40);(9,12,14,23)(10,17,19,21)(11,18,20,22)(13,15,24,16)(25,28,30,38)(26,33,32,39)(27,37,35,29)(31,40,36,34);(9,13)(10,18)(11,21)(12,15)(14,24)(16,23)(17,20)(19,22)(25,29,30,37)(26,34,32,40)(27,38,35,28)(31,39,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842073);