#468772

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

742 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)(9,10,14,19,15,20,13,16)(11,21,24,33,25,31,23,17)(12,18,29,34,26,36,28,30)(22,37,39,32,38,35,27,40);(2,5)(6,8)(10,16,20,19)(12,26)(17,31,33,21)(18,34,36,30)(22,38)(27,39)(28,29)(32,35,40,37);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,21)(18,35)(19,31)(20,33)(22,28)(26,39)(29,38)(30,40)(32,34)(36,37);(3,7)(6,8)(9,12,14,29,15,26,13,28)(10,18,19,34,20,36,16,30)(11,22,24,39,25,38,23,27)(17,32,21,35,33,40,31,37)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efg,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^b=di,d^c=de,e^c=ei,f^c=fi,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, 375968);