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 | #59029 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,22,24,21)(25,26,30,34)(27,35,36,31)(28,32,39,37)(29,33,38,40);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,21)(20,24)(22,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,39)(35,36)(38,40);(1,3)(2,4)(5,8)(6,7)(12,20)(16,23)(19,24)(21,22)(25,27,30,36)(26,31,34,35)(28,38,39,29)(32,33,37,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,21)(17,24)(18,23)(25,28,30,39)(26,32,34,37)(27,29,36,38)(31,40,35,33);(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,40)(30,34)(33,38)(35,36)(37,39);(9,13)(10,17)(11,18)(12,21)(14,15)(16,19)(20,22)(23,24)(25,29)(26,33)(27,28)(30,38)(31,37)(32,35)(34,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8641602);