#340519

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247715);