#6996807

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

2228 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)(9,10,14,21,16,20,15,32)(11,22,25,37,24,36,30,17)(12,26,28,18,29,39,33,35)(13,19,23,38,31,40,27,34);(1,2)(3,7)(4,6)(5,8)(9,11,16,24)(10,17,20,37)(12,27,29,23)(13,28,31,33)(14,30,15,25)(18,38,35,34)(19,26,40,39)(21,36,32,22);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,23)(13,30)(14,28)(15,33)(16,29)(17,34)(19,22)(20,35)(21,39)(24,27)(25,31)(26,32)(36,40)(37,38);(3,8)(5,7)(9,13,14,23,16,31,15,27)(10,19,21,38,20,40,32,34)(11,12,25,28,24,29,30,33)(17,35,22,26,37,18,36,39);(2,6)(5,7)(10,20)(11,24)(13,31)(18,35)(21,32)(23,27)(25,30)(26,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,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, 6904003);