#329513

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29662	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

716 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)(9,10,12,16,13,17,14,18)(11,19,22,24,21,15,20,23)(25,26,28,32,29,33,37,31)(27,30,34,39,35,38,36,40);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24)(25,27)(26,30)(28,34)(29,35)(31,40)(32,39)(33,38)(36,37);(1,3)(2,6)(4,7)(5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(26,31)(27,34)(28,37)(32,33)(35,36)(39,40);(3,7)(6,8)(9,12,13,14)(10,16,17,18)(11,20,21,22)(15,24,19,23)(27,35)(30,38)(34,36)(39,40)

Polycyclic presentation:

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

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, 236709);