#345878

order	512
nilpotency class	3
isoclinism	isoclinic class of #345848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253074);