Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
956 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,23)(13,17)(14,18)(15,21)(16,27)(20,34)(22,33)(24,38)(25,30)(26,37)(28,39)(29,35)(31,40)(32,36);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,24)(13,21)(14,22)(16,28)(17,19)(18,29)(20,26)(23,34)(25,35)(27,36)(30,33)(31,32)(37,38)(39,40);(1,3,5,4)(2,7,8,6)(9,12,25,40)(10,16,30,37)(11,20,35,39)(13,26,14,27)(15,28,33,34)(17,31,18,23)(19,32,29,38)(21,24,22,36);(1,4,5,3)(2,7,8,6)(10,17)(11,21)(16,31)(18,30)(20,24)(22,35)(23,37)(36,39)Polycyclic presentation:
2^9|b^2=ei,c^2=ef,d^2=ei,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 | #72317534, #72317576, #72317625, … |
| maximal normal subgroup of | #72317534, #72317576, #72317625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410683);