#357369

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264565);