#178022

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

793 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85218);