#106725

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
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

859 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,f^b=fhi,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, 13921);