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 | |
4305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,16,13)(11,19,15,17)(12,21,14,18)(20,23,22,24)(28,36)(31,38)(35,40)(37,39);(1,3,6,4)(2,7,5,8)(9,11,16,15)(10,17,13,19)(12,20,14,22)(18,23,21,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(9,12,16,14)(10,18,13,21)(11,20,15,22)(17,23,19,24)(25,27)(26,29)(28,35)(30,33)(31,37)(32,34)(36,40)(38,39);(9,13,16,10)(11,19,15,17)(12,18,14,21)(20,23,22,24)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,40)(34,38);(1,4,6,3)(2,7,5,8)(9,14,16,12)(10,18,13,21)(11,20,15,22)(17,24,19,23)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,40)(37,38);(1,5,6,2)(3,8,4,7)(9,15,16,11)(10,19,13,17)(12,22,14,20)(18,24,21,23)(25,29,30,32)(26,33,34,27)(28,37,36,39)(31,40,38,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,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.
| maximal normal subgroup of | #72219281, #72219283 |
| maximal quotient of | #72219271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619995);