#357210

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=gh,g^2=i,b^a=be,c^a=cf,d^b=de

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264406);