#498626

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₁₆
socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,g^2=h,h^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, 405822);