#6996399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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

2309 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)(9,10,14,19,16,12,15,26)(11,20,23,21,24,33,31,17)(13,18,28,35,30,25,29,39)(22,34,27,36,37,38,40,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,21)(13,27)(14,31)(15,23)(16,24)(18,34)(19,33)(20,26)(22,28)(25,38)(29,37)(30,40)(32,35)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12,16,10)(11,21,24,17)(13,25,30,18)(14,26,15,19)(20,23,33,31)(22,36,37,32)(27,38,40,34)(28,39,29,35);(3,8)(5,7)(9,13,15,29,16,30,14,28)(10,18,26,39,12,25,19,35)(11,22,31,40,24,37,23,27)(17,32,33,38,21,36,20,34);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=f,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

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, 6903595);