#178408

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgi,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85604);