#447858

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
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

635 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=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, 355054);