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 | |
4621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,17,19,12)(13,21,23,16)(18,20,22,24)(25,26)(27,31)(28,33)(29,38)(30,32)(34,39)(35,36)(37,40);(3,7)(6,8)(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(25,27)(26,31)(28,29)(30,35)(32,36)(33,38)(34,40)(37,39);(1,2,5,4)(3,6,7,8)(10,15)(12,17)(16,21)(20,24)(26,32)(27,35)(28,37)(34,38);(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28,30,37)(26,33,32,40)(27,29,35,39)(31,38,36,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,40)(38,39);(1,4,5,2)(3,8,7,6)(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,29)(26,34)(27,37)(28,35)(30,39)(31,33)(32,38)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725628);