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 | #37440 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,3,5,8)(2,6,4,7)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(1,4,5,2)(3,7,8,6)(10,15)(11,18)(17,20)(19,24)(26,31)(27,33)(29,37)(35,40);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=g,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7670644);