#6439760

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437608
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346956);