#106354

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

962 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13550);