#451805

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

787 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=e,d^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, 359001);