#357203

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264399);