#110225

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 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:

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

Polycyclic presentation:

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

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, 17421);