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 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26,30,32)(27,35,37,31)(28,36,38,33)(29,34,39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,34)(29,36)(30,37)(32,35)(33,39)(38,40);(1,2,5,4)(3,6,8,7)(11,17)(15,18)(19,23)(22,24)(26,32)(28,38)(31,35)(34,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,33)(27,34)(29,35)(30,38)(31,39)(32,36)(37,40);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29,30,39)(26,34,32,40)(27,36,37,33)(28,35,38,31);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,27)(26,35)(28,34)(29,33)(30,37)(31,32)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7570337);