#346066

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ef,c^2=e,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253262);