#468487

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

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

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, 375683);