#462838

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

835 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)(9,10,13,15,14,19,28,18)(11,20,22,29,23,16,27,31)(12,17,26,30,24,35,39,33)(21,32,25,40,37,36,38,34);(1,3,5,4)(2,6,7,8)(10,15,19,18)(12,24)(16,31,20,29)(17,33,35,30)(21,37)(25,38)(26,39)(32,34,36,40);(2,7)(6,8)(9,11)(10,16)(12,25)(13,27)(14,23)(15,29)(17,34)(18,31)(19,20)(21,26)(22,28)(24,38)(30,36)(32,33)(35,40)(37,39);(1,4,5,3)(2,8,7,6)(9,12,13,26,14,24,28,39)(10,17,15,30,19,35,18,33)(11,21,22,25,23,37,27,38)(16,32,31,40,20,36,29,34)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,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^b=di,d^c=de,e^c=ei,h^c=hi

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, 370034);