#477171

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384367);