#7021364

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6928560);