#471043

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

638 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)(3,6)(4,5)(7,8)(9,10,16,20,15,21,14,30)(11,22,13,19,25,37,24,17)(12,18,31,34,28,35,27,38)(23,39,40,32,26,33,29,36);(9,11,14,24,15,25,16,13)(10,17,20,22,21,19,30,37)(12,26,31,29,28,23,27,40)(18,32,38,39,35,36,34,33);(2,5)(6,8)(9,12)(10,18)(11,23)(13,29)(14,31)(15,28)(16,27)(17,32)(19,36)(20,38)(21,35)(22,39)(24,40)(25,26)(30,34)(33,37);(1,3,4,7)(2,6,5,8)(9,13,16,25,15,24,14,11)(10,19,20,37,21,17,30,22)(12,23,31,40,28,26,27,29)(18,33,34,36,35,39,38,32)

Polycyclic presentation:

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

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, 378239);