#280845

order	512
nilpotency class	3
isoclinism	isoclinic class of #280843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

509 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188041);