#460614

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1000 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26,28,17)(13,19)(14,18)(15,20)(16,22)(23,35,38,40)(24,32)(25,31)(27,33,29,34)(30,37,39,36);(1,3,4,7)(2,5,6,8)(9,11,13,22,15,25,14,24)(10,16,18,21,20,32,19,31)(12,23,29,36,28,38,27,37)(17,30,34,35,26,39,33,40);(2,6)(5,8)(11,22)(12,27)(13,14)(16,21)(17,33)(18,19)(24,25)(26,34)(28,29)(31,32)(35,40)(36,37);(9,12)(10,17)(11,23)(13,29)(14,27)(15,28)(16,30)(18,34)(19,33)(20,26)(21,35)(22,36)(24,37)(25,38)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,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, 367810);