#6452027

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437793
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2627 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)(3,8)(4,6)(5,7)(9,10,13,19,15,18,14,20)(11,21,23,31,25,32,24,16)(12,17,27,35,28,34,29,36)(22,33,26,39,37,40,38,30);(9,11,15,25)(10,16,18,31)(12,26,28,38)(13,24,14,23)(17,33,34,40)(19,32,20,21)(22,29,37,27)(30,36,39,35);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,35)(20,36)(21,33)(23,26)(24,38)(25,37)(31,39)(32,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,35)(20,36)(21,33)(23,26)(24,38)(25,37)(31,39)(32,40);(10,18)(16,31)(17,34)(19,20)(21,32)(30,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6359223);