#357003

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,4)(5,6,8,7)(9,10,14,20,15,21,32,19)(11,22,24,16,25,34,40,35)(12,26,28,33,29,39,27,17)(13,18,31,36,23,37,30,38);(1,2)(3,4)(9,11,14,24,15,25,32,40)(10,16,19,22,21,35,20,34)(12,23,27,31,29,13,28,30)(17,18,26,36,33,37,39,38);(5,6,8,7)(9,12,15,29)(10,17,21,33)(11,23,25,13)(14,27,32,28)(16,18,35,37)(19,26,20,39)(22,36,34,38)(24,31,40,30);(2,4)(5,7,8,6)(9,13)(10,18)(11,12)(14,31)(15,23)(16,33)(17,35)(19,38)(20,36)(21,37)(22,26)(24,28)(25,29)(27,40)(30,32)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fgi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,f^c=fi,g^c=gi,e^d=ei,h^d=hi

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, 264199);