#166254

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73450);