#6439768

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

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

Polycyclic presentation:

2^9|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, 6346964);