#453865

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = 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	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

575 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)(6,8)(9,10,14,17,15,20,13,19)(11,21,24,16,25,31,23,33)(12,26,29,18,30,34,28,36)(22,32,37,39,38,40,27,35);(1,2,4,5)(3,6,7,8)(9,11,14,23,15,25,13,24)(10,16,17,21,20,33,19,31)(12,27,29,38,30,37,28,22)(18,35,34,40,36,39,26,32);(1,3,4,7)(2,6,5,8)(10,17,20,19)(16,31,33,21)(18,34,36,26)(32,39,40,35);(2,5)(6,8)(9,12,14,29,15,30,13,28)(10,18,19,26,20,36,17,34)(11,22,24,37,25,38,23,27)(16,32,21,35,33,40,31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361061);