#433080

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 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)(3,5)(4,6)(7,8)(9,10,15,20,16,21,14,32)(11,22,24,17,25,35,27,36)(12,26,28,39,29,38,33,18)(13,19,30,34,31,37,23,40);(1,3,4,7)(2,5,6,8)(9,11,15,24,16,25,14,27)(10,17,20,35,21,36,32,22)(12,13,28,30,29,31,33,23)(18,37,26,40,39,19,38,34);(2,6)(5,8)(9,12)(10,18)(11,23)(13,27)(14,28)(15,33)(16,29)(17,34)(19,35)(20,38)(21,39)(22,37)(24,31)(25,30)(26,32)(36,40);(9,13,15,30,16,31,14,23)(10,19,32,40,21,37,20,34)(11,12,27,33,25,29,24,28)(17,18,35,26,36,39,22,38)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,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, 340276);