Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| 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 | |
814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,18)(11,15,20,28)(12,22,25,37)(13,17,26,32)(16,30,31,24)(19,33,23,36)(21,29,35,40)(27,38,39,34);(2,5)(6,7)(9,11,14,20)(10,15,18,28)(12,23,25,19)(13,21,26,35)(16,27,31,39)(17,29,32,40)(22,36,37,33)(24,34,30,38);(9,12,13,24)(10,16,17,22)(11,19,21,34)(14,25,26,30)(15,27,29,33)(18,31,32,37)(20,23,35,38)(28,39,40,36);(1,3,4,8)(2,6,5,7)(11,20)(15,28)(19,23)(21,35)(27,39)(29,40)(33,36)(34,38)Polycyclic presentation:
2^9|a^2=ei,b^2=i,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72317725, #72317746, #72317806, … |
| maximal normal subgroup of | #72317725, #72317746, #72317806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410204);