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 | #38092 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,16,19,22)(12,17,21,24)(13,20,23,18)(28,36)(32,39)(34,40)(37,38);(1,3,5,8)(2,6,7,4)(10,15)(12,21)(16,22)(18,20)(25,26)(27,29)(28,32)(30,33)(31,35)(34,37)(36,39)(38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,19)(17,24)(18,20)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39);(1,4)(2,3)(5,6)(7,8)(9,11,14,19)(10,16,15,22)(12,20,21,18)(13,17,23,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,12,14,21)(10,17,15,24)(11,20,19,18)(13,16,23,22)(25,29,30,31)(26,27,33,35)(28,37,36,38)(32,34,39,40);(9,13,14,23)(10,18,15,20)(11,17,19,24)(12,22,21,16)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9624966);