#6444129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,25)(13,18)(14,20)(15,21)(17,23)(19,30)(22,31)(24,32)(26,34)(27,37)(28,35)(29,36)(33,38)(39,40);(2,6)(7,8)(9,11)(10,16)(12,14)(13,26)(15,23)(17,21)(18,34)(19,32)(20,25)(22,28)(24,30)(27,38)(29,39)(31,35)(33,37)(36,40);(1,3,5,4)(2,7,6,8)(9,12,14,23,15,24,30,11)(10,17,19,25,21,16,20,32)(13,26,27,38,29,39,28,22)(18,33,35,34,36,31,37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,22)(12,26)(14,27)(15,29)(16,31)(17,33)(19,35)(20,37)(21,36)(23,38)(24,39)(25,34)(28,30)(32,40);(11,23)(12,24)(16,17)(22,38)(25,32)(26,39)(31,33)(34,40)

Polycyclic presentation:

2^9|c^2=fg,d^2=f,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,e^c=ei,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, 6351325);