#460787

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1083 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)(9,10,13,15,14,19,29,18)(11,20,22,30,23,16,28,32)(12,24,26,36,27,17,34,31)(21,37,25,35,38,33,40,39);(1,3,4,7)(2,5,6,8)(10,15,19,18)(16,32,20,30)(17,31,24,36)(33,39,37,35);(2,6)(5,8)(9,11)(10,16)(12,25)(13,28)(14,23)(15,30)(17,35)(18,32)(19,20)(21,26)(22,29)(24,39)(27,40)(31,37)(33,36)(34,38);(9,12,13,26,14,27,29,34)(10,17,18,36,19,24,15,31)(11,21,22,25,23,38,28,40)(16,33,30,35,20,37,32,39)

Polycyclic presentation:

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

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, 367983);