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 | #38364 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,18,12,16)(13,22,23,19)(14,20,24,21)(25,26)(27,35)(28,34)(29,32)(30,36)(31,33)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,18)(13,20,23,21)(14,19,24,22)(25,27,31,29)(26,32,33,35)(28,37,36,39)(30,38,34,40);(10,17)(11,12)(19,22)(20,21)(26,33)(28,36)(32,35)(37,39);(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,22,24,19)(25,28,31,36)(26,34,33,30)(27,37,29,39)(32,40,35,38);(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,16,24,18)(25,29,31,27)(26,35,33,32)(28,39,36,37)(30,40,34,38);(1,3,4,7)(2,6,5,8)(9,14)(10,20)(11,19)(12,22)(13,18)(15,24)(16,23)(17,21)(25,30,31,34)(26,36,33,28)(27,38,29,40)(32,39,35,37)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7827406);