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 | #45522 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,17,19)(12,16,22,24)(18,23,20,21)(25,26,31,34)(27,35,33,29)(28,38,30,32)(36,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,20)(14,17)(15,19)(16,21)(18,22)(23,24)(25,27,31,33)(26,29,34,35)(28,36,30,37)(32,40,38,39);(2,3)(6,7)(10,15)(11,17)(16,24)(18,20)(25,28,31,30)(26,32,34,38)(27,36,33,37)(29,40,35,39);(4,8)(6,7)(9,12,14,22)(10,16,15,24)(11,18,17,20)(13,23,19,21)(25,26)(27,35)(28,32)(29,33)(30,38)(31,34)(36,39)(37,40);(1,3,5,2)(4,7,8,6)(9,10)(11,19)(12,16)(13,17)(14,15)(18,21)(20,23)(22,24)(25,29,31,35)(26,33,34,27)(28,39,30,40)(32,36,38,37);(1,4,5,8)(2,6,3,7)(9,13,14,19)(10,17,15,11)(12,21,22,23)(16,18,24,20)(25,30,31,28)(26,32,34,38)(27,37,33,36)(29,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8558595);