#173868

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

799 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)(4,7)(5,8)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19)(26,31)(28,29)(32,39)(33,36)(34,37)(38,40);(2,6)(3,4)(7,8)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17)(25,26)(27,32)(28,31)(29,36)(30,33)(34,38)(35,40)(37,39);(2,7)(3,4)(6,8)(25,27)(26,32)(28,34)(29,37)(30,35)(31,38)(33,40)(36,39);(2,8)(6,7)(11,18)(14,23)(17,24)(19,21)(25,28,30,29)(26,31,33,36)(27,34,35,37)(32,38,40,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81064);