#149381

order	512
nilpotency class	4
isoclinism	isoclinic class of #140781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56577);