#6449009

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	#23439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2232 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)(9,10,14,20,15,21,16,19)(11,22,26,34,27,35,25,17)(12,28,30,37,13,18,32,38)(23,39,29,33,24,36,31,40);(9,11)(10,17)(12,29)(13,31)(14,25)(15,27)(16,26)(18,36)(19,22)(20,35)(21,34)(23,30)(24,32)(28,39)(33,38)(37,40);(2,5)(6,8)(9,12,15,13)(10,18,21,28)(11,23,27,24)(14,30,16,32)(17,33,34,40)(19,37,20,38)(22,36,35,39)(25,31,26,29);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,21,28)(11,24,27,23)(14,32,16,30)(17,33,34,40)(19,37,20,38)(22,36,35,39)(25,29,26,31);(9,14,15,16)(10,19,21,20)(11,25,27,26)(12,30,13,32)(17,22,34,35)(18,37,28,38)(23,31,24,29)(33,36,40,39)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=g,e^2=i,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, 6356205);