#6598312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(13,14,19,24)(15,25,29,36)(16,30,33,39)(17,22,18,23)(20,28,37,27)(21,34,40,32)(26,35,31,38)(41,43,44,42);(5,6,8,9)(7,10,11,12)(13,15,18,28,19,29,17,27)(14,20,22,36,24,37,23,25)(16,31,32,43,33,26,34,42)(21,38,30,41,40,35,39,44);(2,4)(6,9)(7,11)(13,16,17,34,19,33,18,32)(14,21,23,39,24,40,22,30)(15,26,27,43,29,31,28,42)(20,35,25,41,37,38,36,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,33,34)(20,36,37,25)(21,39,40,30)(26,42,31,43)(35,41,38,44);(6,9)(7,11)(16,33)(21,40)(26,31)(30,39)(32,34)(35,38)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505508);