#6996813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
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

2242 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)(9,10,14,19,16,20,15,31)(11,21,23,33,12,24,26,17)(13,18,29,36,28,35,30,40)(22,34,27,39,25,37,38,32);(1,2)(3,7)(4,6)(5,8)(9,11,16,12)(10,17,20,33)(13,27,28,38)(14,26,15,23)(18,34,35,37)(19,24,31,21)(22,30,25,29)(32,40,39,36);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,17,20,33)(13,25,28,22)(14,26,15,23)(18,32,35,39)(19,21,31,24)(27,29,38,30)(34,40,37,36);(3,8)(5,7)(9,13,15,30,16,28,14,29)(10,18,31,40,20,35,19,36)(11,22,26,38,12,25,23,27)(17,32,24,37,33,39,21,34);(2,6)(5,7)(13,28)(18,35)(22,25)(27,38)(29,30)(32,39)(34,37)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,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,g^c=gi,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, 6904009);