#6439738

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 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:

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

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,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, 6346934);