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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(10,11)(12,19)(16,18)(20,21)(28,36)(31,39)(35,40)(37,38);(1,2,4,5)(3,6,7,8)(10,11)(15,17)(16,18)(23,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24)(25,27)(26,29)(28,35)(30,33)(31,38)(32,34)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,19,15)(13,16,22,18)(20,23,21,24)(25,28,30,36)(26,31,34,39)(27,35,33,40)(29,38,32,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,27)(26,32)(28,35)(29,34)(30,33)(31,37)(36,40)(38,39);(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,29,30,32)(26,33,34,27)(28,37,36,38)(31,35,39,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595473);