#345593

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252789);