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 | #39429 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,12,17,20)(13,16,22,23)(18,19,24,21)(25,26,32,30)(27,36,39,33)(28,29,34,38)(31,35,40,37);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27,32,39)(26,33,30,36)(28,35,34,37)(29,31,38,40);(10,15)(11,17)(16,23)(18,24)(25,28)(26,29)(27,37)(30,38)(31,36)(32,34)(33,40)(35,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,29,32,38)(26,34,30,28)(27,31,39,40)(33,37,36,35);(1,4,5,3)(2,8,6,7)(9,12)(10,11)(13,21)(14,20)(15,17)(16,24)(18,23)(19,22)(25,30,32,26)(27,36,39,33)(28,38,34,29)(31,35,40,37);(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,31)(26,35)(27,38)(28,36)(29,39)(30,37)(32,40)(33,34)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7765093);