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×#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 | |
828 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)(11,19)(12,23)(13,17)(14,18)(15,21)(16,26)(20,34)(22,33)(24,38)(25,37)(27,31)(28,39)(29,36)(30,40)(32,35);(2,5)(6,7)(9,11)(10,15)(12,24)(13,21)(14,22)(16,28)(17,19)(18,29)(20,25)(23,34)(26,35)(27,36)(30,32)(31,33)(37,38)(39,40);(9,12,14,26)(10,16,18,23)(11,20,22,35)(13,25,27,40)(15,28,29,38)(17,30,31,37)(19,32,33,34)(21,24,36,39);(1,3,4,8)(2,6,5,7)(10,17)(11,21)(16,30)(18,31)(20,24)(22,36)(23,37)(35,39)Polycyclic presentation:
2^9|a^2=ei,c^2=f,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, #72317587, #72317623, … |
| maximal normal subgroup of | #72317534, #72317587, #72317623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410203);