#106370

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh

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, 13566);