#6439734

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

2115 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)(9,10,13,19)(11,21,24,16)(12,18,27,35)(14,17,15,20)(22,32,25,31)(23,37,38,30)(26,40,39,34)(28,33,29,36);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,26)(13,24)(14,25)(15,22)(17,32)(18,34)(19,21)(20,31)(23,28)(27,39)(29,38)(30,33)(35,40)(36,37);(10,17)(11,22)(14,15)(18,33)(19,20)(23,26)(24,25)(28,29)(31,32)(34,40)(35,36)(38,39);(1,4,5,3)(2,6,7,8)(9,12,13,27)(10,18,19,35)(11,23,24,38)(14,28,15,29)(16,30,21,37)(17,33,20,36)(22,26,25,39)(31,34,32,40);(10,19)(16,21)(17,20)(18,35)(30,37)(31,32)(33,36)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,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, 6346930);