#113600

order	512
nilpotency class	3
isoclinism	isoclinic class of #10192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

935 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20796);