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 | #51423 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4235 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)(9,10,14,16)(11,15,19,24)(12,17,18,13)(20,22,23,21);(1,3,5,8)(2,6,4,7)(9,11,14,19)(10,15,16,24)(12,20,18,23)(13,21,17,22)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,2,5,4)(3,6,8,7)(10,16)(11,19)(12,18)(21,22)(26,31)(29,38)(32,39)(35,40);(9,12)(10,17)(11,20)(13,16)(14,18)(15,22)(19,23)(21,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(1,2,5,4)(3,6,8,7)(9,13,14,17)(10,18,16,12)(11,21,19,22)(15,23,24,20)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,19)(16,24)(17,22)(18,23)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7733704);