#479775

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386971);