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 | #47870 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24);(1,2)(3,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26,31,30)(27,35,37,32)(28,33,36,38)(29,34,40,39);(10,15)(11,17)(13,21)(19,24)(25,27,31,37)(26,32,30,35)(28,29,36,40)(33,39,38,34);(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,28)(26,33)(27,29)(30,38)(31,36)(32,39)(34,35)(37,40);(1,3,5,4)(2,6,8,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29)(26,34)(27,36)(28,37)(30,39)(31,40)(32,33)(35,38);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,26)(27,32,37,35)(28,38,36,33)(29,39,40,34)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7689918);