#7021387

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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

2147 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)(9,10,14,20,15,19,27,21)(11,13,18,32,23,33,24,16)(12,25,30,17,29,37,40,38)(22,35,39,28,26,34,36,31);(1,2)(3,7)(4,6)(5,8)(9,11,15,23)(10,16,19,32)(12,26,29,22)(13,20,33,21)(14,24,27,18)(17,35,38,34)(25,28,37,31)(30,39,40,36);(1,3)(2,5)(4,8)(6,7)(9,12,27,40,15,29,14,30)(10,17,20,37,19,38,21,25)(11,22,18,39,23,26,24,36)(13,31,16,34,33,28,32,35);(3,8)(5,7)(9,13,14,32,15,33,27,16)(10,18,20,23,19,24,21,11)(12,28,30,34,29,31,40,35)(17,36,37,22,38,39,25,26);(2,6)(5,7)(10,19)(11,23)(12,29)(18,24)(20,21)(28,31)(30,40)(34,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,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

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, 6928583);