Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,24)(13,18)(14,19)(15,20)(17,26)(21,30)(22,31)(23,32)(25,39)(27,34)(28,35)(29,36)(33,37)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,21)(14,22)(15,23)(17,33)(18,30)(19,31)(20,32)(24,39)(26,37)(27,36)(28,38)(29,34)(35,40);(9,12,15,13)(10,17,20,34)(11,21,23,25)(14,26,28,27)(16,29,32,33)(18,19,24,35)(22,36,38,37)(30,40,39,31);(9,13,15,12)(10,18,20,24)(11,21,23,25)(14,26,28,27)(16,30,32,39)(17,19,34,35)(22,37,38,36)(29,40,33,31)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72281820, #72281905, #72281983, … |
| maximal normal subgroup of | #72281820, #72281905, #72281983, … |
| maximal quotient of | #72282608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419229);