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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,13,17)(11,15,19,14)(12,16,20,23)(18,22,24,21)(25,26)(27,33)(28,38)(29,34)(30,32)(31,35)(36,40)(37,39);(1,3,5,8)(2,6,7,4)(9,11,13,19)(10,14,17,15)(12,18,20,24)(16,21,23,22)(25,27,32,35)(26,33,30,31)(28,37,29,36)(34,40,38,39);(1,3)(2,4)(5,8)(6,7)(12,20)(16,23)(18,24)(21,22)(25,28,32,29)(26,34,30,38)(27,36,35,37)(31,40,33,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,29,32,28)(26,34,30,38)(27,37,35,36)(31,40,33,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30,32,26)(27,33,35,31)(28,38,29,34)(36,39,37,40);(9,12,13,20)(10,16,17,23)(11,18,19,24)(14,21,15,22)(25,31,32,33)(26,35,30,27)(28,39,29,40)(34,36,38,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,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, 8279454);