#345873

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	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

585 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)(11,20)(12,26)(13,16)(14,33)(15,19)(17,23)(18,30)(21,34)(22,25)(24,32)(27,35)(28,29)(31,36)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(9,11,14,23,16,25,15,24)(10,17,19,20,13,32,33,22)(12,27,30,21,31,38,29,39)(18,35,26,40,28,37,36,34);(2,5)(3,7)(6,8)(9,12)(10,18)(11,21)(13,28)(14,29)(15,30)(16,31)(17,34)(19,36)(20,37)(22,35)(23,27)(24,38)(25,39)(26,33)(32,40);(3,8)(6,7)(9,13,16,10)(11,22,25,20)(12,28,31,18)(14,19,15,33)(17,23,32,24)(21,35,39,37)(26,29,36,30)(27,40,38,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,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, 253069);