#453800

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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)(3,7)(4,6)(5,8)(9,10,13,15,14,19,21,18)(11,16,20,31,24,32,23,29)(12,25,27,33,28,34,26,17)(22,36,35,40,37,39,38,30);(1,3)(2,5)(4,8)(6,7)(10,15,19,18)(11,20,24,23)(12,26,28,27)(17,33)(22,37)(25,34)(30,39,40,36)(35,38);(2,6)(5,7)(9,11,21,20,14,24,13,23)(10,16,18,31,19,32,15,29)(12,22,26,35,28,37,27,38)(17,30,34,36,33,40,25,39);(3,8)(5,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,25)(16,30)(18,34)(19,33)(20,35)(21,26)(23,38)(24,37)(29,39)(31,36)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360996);