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 | |
4593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,36)(28,38)(29,35)(30,32)(31,33)(34,39)(37,40);(3,4)(6,8)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27,31,30)(26,32,33,36)(28,37,39,35)(29,34,40,38);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(16,21)(19,24)(26,33)(28,39)(32,36)(35,37);(3,4)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28)(26,34)(27,37)(29,36)(30,35)(31,39)(32,40)(33,38);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,40)(26,35,33,37)(27,38,30,34)(28,32,39,36);(1,2)(3,6)(4,8)(5,7)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30,31,27)(26,36,33,32)(28,35,39,37)(29,38,40,34)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10054250);