#6575181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2149 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)(5,6)(7,10)(8,9)(11,12)(15,23)(16,28)(19,34)(20,37)(24,35)(25,27)(29,38)(30,33);(6,9)(10,12)(13,14,17,21)(15,24,23,35)(16,29,28,38)(18,32,31,22)(19,25,34,27)(20,33,37,30)(26,36,39,43)(40,42,44,41);(2,4)(13,15,17,23)(14,19,21,34)(16,26,28,39)(18,25,31,27)(20,36,37,43)(22,35,32,24)(29,40,38,44)(30,42,33,41);(5,7,8,11)(6,10,9,12)(15,25)(16,30)(18,31)(19,35)(20,29)(22,32)(23,27)(24,34)(26,39)(28,33)(36,43)(37,38);(13,16,17,28)(14,20,21,37)(15,26,23,39)(18,33,31,30)(19,36,34,43)(22,38,32,29)(24,40,35,44)(25,41,27,42)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6482377);