#475046

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1061 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)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,21,23,33,25,32,24,16)(12,26,28,34,30,17,29,27)(22,36,39,37,40,35,38,31);(2,5)(6,8)(9,11)(10,16,20,33)(12,22)(13,23)(14,24)(15,25)(17,31,26,37)(18,21,19,32)(27,35,34,36)(28,38)(29,39)(30,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,21)(12,27)(13,19)(14,18)(15,20)(16,23)(17,28)(22,37)(24,33)(25,32)(26,29)(30,34)(31,40)(35,38)(36,39);(3,7)(6,8)(9,12,14,29,15,30,13,28)(10,17,19,34,20,26,18,27)(11,22,24,39,25,40,23,38)(16,31,32,36,33,37,21,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382242);