#207463

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2619	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114659);