#446281

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,5,4)(3,7,8,6)(9,10,15,18)(11,13,16,14)(12,20)(17,23)(19,22)(21,24);(1,3,5,8)(2,6,4,7)(10,16)(11,18)(12,21)(13,14)(17,20)(22,23);(1,2,5,4)(3,6,8,7)(9,11,14,10,15,16,13,18)(12,19,21,17,23,24,22,20);(1,4,5,2)(3,6,8,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,24)(18,20)

Polycyclic presentation:

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

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, 353477);