#6439258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
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

2070 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)(9,10,15,20)(11,22,25,16)(12,27,30,35)(13,18,23,38)(14,19,33,21)(17,29,37,31)(24,40,26,36)(28,34,32,39);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,16,20,22)(12,28,30,32)(13,31,23,29)(14,26,33,24)(17,18,37,38)(19,40,21,36)(27,39,35,34);(1,4,5,3)(2,6,7,8)(9,12,15,30)(10,17,20,37)(11,23,25,13)(14,29,33,31)(16,34,22,39)(18,36,38,40)(19,27,21,35)(24,32,26,28);(9,13)(10,18)(11,12)(14,28)(15,23)(16,35)(17,36)(19,34)(20,38)(21,39)(22,27)(24,29)(25,30)(26,31)(32,33)(37,40);(9,14,15,33)(10,19,20,21)(11,24,25,26)(12,29,30,31)(13,28,23,32)(16,36,22,40)(17,27,37,35)(18,34,38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,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, 6346454);