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 | #2670 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(13,21)(16,20)(18,24)(22,23)(25,26,29,32)(27,33,34,30)(28,31,37,40)(35,39,36,38);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,21)(23,24)(27,34)(30,33)(35,36)(38,39);(10,12)(15,17)(16,20)(22,23)(25,27,29,34)(26,30,32,33)(28,36,37,35)(31,39,40,38);(1,3,5,4)(2,6,8,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,29,37)(26,31,32,40)(27,35,34,36)(30,38,33,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,37)(39,40);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626524);