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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,20,19)(12,17,23,22)(13,18,24,21);(2,3)(6,7)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,23)(22,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,5,2)(4,7,8,6)(10,16)(11,20)(17,22)(18,21)(26,31)(29,38)(32,39)(36,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,21)(13,15)(14,23)(16,22)(18,20)(19,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(9,13,14,24)(10,18,16,21)(11,22,20,17)(12,19,23,15)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,24)(18,23)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8809708);