#356723

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=g,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, 263919);