#451623

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

770 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=ef,e^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,h^c=hi,e^d=ei,f^d=fi

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, 358819);