#6439774

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2067 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)(11,22)(12,26)(13,19)(14,20)(15,21)(16,33)(17,24)(18,30)(23,37)(25,36)(27,40)(28,38)(29,35)(31,34)(32,39);(2,5)(6,8)(9,11)(10,17)(12,27)(13,28)(14,23)(15,24)(16,25)(18,19)(20,36)(21,22)(26,34)(29,31)(30,32)(33,37)(35,40)(38,39);(9,12,15,29)(10,18,21,38)(11,13,24,32)(14,30,16,28)(17,34,22,40)(19,37,39,36)(20,35,33,26)(23,27,25,31);(9,13,15,32)(10,19,21,39)(11,12,24,29)(14,31,16,27)(17,35,22,26)(18,37,38,36)(20,34,33,40)(23,28,25,30);(1,3,4,7)(2,6,5,8)(9,14,15,16)(10,20,21,33)(11,23,24,25)(12,28,29,30)(13,31,32,27)(17,36,22,37)(18,26,38,35)(19,34,39,40)

Polycyclic presentation:

2^9|c^2=i,d^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,e^c=ei,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, 6346970);