#6439623

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
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2259 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,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, 6346819);