#462769

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

744 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)(4,8)(5,7)(9,10,13,16,15,20,14,19)(11,21,23,17,25,31,24,33)(12,18,27,30,29,36,28,35)(22,37,26,40,39,34,38,32);(1,3,5,4)(2,6,7,8)(10,16,20,19)(17,31,33,21)(18,30,36,35)(32,37,40,34);(2,7)(6,8)(9,11)(10,17)(12,26)(13,24)(14,23)(15,25)(16,21)(18,34)(19,31)(20,33)(22,27)(28,39)(29,38)(30,40)(32,35)(36,37);(1,4,5,3)(2,8,7,6)(9,12,14,28,15,29,13,27)(10,18,19,35,20,36,16,30)(11,22,24,38,25,39,23,26)(17,32,21,34,33,40,31,37)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 369965);