#6578438

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485634);