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 | #57285 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,18)(11,14,13,17)(12,21,23,16)(19,22,20,24)(25,26)(27,35)(28,34)(29,33)(30,31)(32,36)(37,39)(38,40);(2,3)(6,7)(9,11,15,13)(10,14,18,17)(12,20,23,19)(16,22,21,24)(26,31)(27,36)(33,34)(37,38);(1,3,5,2)(4,7,8,6)(9,12,15,23)(10,16,18,21)(11,19,13,20)(14,24,17,22)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,20)(13,19)(14,22)(15,23)(17,24)(18,21)(25,28,30,29)(26,33,31,34)(27,37,36,38)(32,39,35,40);(9,13,15,11)(10,17,18,14)(12,20,23,19)(16,22,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(9,14,15,17)(10,11,18,13)(12,22,23,24)(16,20,21,19)(25,29,30,28)(26,34,31,33)(27,38,36,37)(32,40,35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839375);