#445464

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^d=ei,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, 352660);