#266779

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,7,4,6)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,24,22,23)(25,26)(27,35)(28,31)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23)(26,31)(27,36)(29,30)(32,35)(37,38)(39,40);(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(25,27,28,37)(26,32,31,39)(29,36,30,38)(33,40,34,35);(1,4,5,3)(2,7,8,6)(10,16)(14,18)(15,17)(23,24)(26,31)(32,39)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 173975);