Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18,15,20,13,19)(11,21,24,22,26,33,25,16)(12,17,30,35,31,28,29,36)(23,37,27,38,39,34,40,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,27)(13,25)(14,24)(15,26)(17,34)(18,21)(19,33)(20,22)(23,29)(28,37)(30,39)(31,40)(32,35)(36,38);(1,3,4,8)(2,5,6,7)(9,10)(11,22)(12,28)(13,18)(14,19)(15,20)(16,26)(17,31)(21,24)(23,32)(25,33)(27,34)(29,36)(30,35)(37,40)(38,39);(3,8)(5,7)(9,12,15,31)(10,17,20,28)(11,23,26,39)(13,29,14,30)(16,32,22,38)(18,35,19,36)(21,37,33,34)(24,27,25,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,24,26,25)(12,29,31,30)(16,33,22,21)(17,35,28,36)(23,27,39,40)(32,34,38,37)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970290);