#6439742

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2125 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,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, 6346938);