#351632

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=dh,d^c=dh,g^d=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, 258828);