#6643226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550422);