#326916

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37541	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

851 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,19,20,12)(14,18,23,17)(15,24,21,22)(25,26,29,33)(27,35,37,31)(28,34,30,32)(36,40,38,39);(2,5)(6,8)(9,11)(10,12)(13,20)(14,21)(15,23)(16,19)(17,22)(18,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,35)(34,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,19)(10,15,16,21)(11,17,20,18)(14,24,23,22)(26,32)(28,30)(31,39)(33,34)(35,40)(36,38);(3,7)(6,8)(10,16)(12,19)(17,18)(22,24)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234112);