Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501132 |
| 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×Q8 | C24 |
| prime radical | C23 | C2×#120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,19)(12,23)(13,25)(15,28)(17,30)(18,26)(20,33)(21,34)(22,35)(29,40)(36,37)(38,39);(1,3,5,4)(2,6,8,7)(9,10)(11,20)(12,17)(13,26)(14,16)(15,21)(18,25)(19,33)(22,36)(23,30)(24,32)(27,31)(28,34)(29,39)(35,37)(38,40);(9,11,12,19,14,22,23,35)(10,15,17,28,16,29,30,40)(13,21,27,38,25,39,24,34)(18,20,32,37,26,36,31,33);(1,4,5,3)(2,7,8,6)(10,16)(15,29)(17,30)(18,26)(20,36)(28,40)(31,32)(33,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72271508 |
| maximal normal subgroup of | #72271508 |
| maximal quotient of | #72271535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410566);