#106706

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1013 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13902);