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 | #59141 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,19)(12,20)(13,17)(14,18)(15,21)(16,22)(23,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,38)(36,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,21)(10,15,18,19)(12,17,22,24)(13,20,23,16)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(9,10,14,18)(11,19,21,15)(12,16,22,20)(13,17,23,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,38)(36,37)(39,40);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,28,30,33)(26,32,35,29)(27,36,38,40)(31,39,34,37);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,17)(13,15)(14,22)(18,20)(19,23)(21,24)(25,29,30,32)(26,33,35,28)(27,37,38,39)(31,40,34,36);(9,13)(10,17)(11,20)(12,19)(14,23)(15,22)(16,21)(18,24)(25,27)(26,34)(28,36)(29,39)(30,38)(31,35)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=gh,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7760542);