#6576863

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2696 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)(7,12)(8,10)(9,11)(15,23)(16,27)(19,34)(20,36)(24,38)(25,33)(28,37)(29,30);(5,7,9,8)(6,10,11,12)(13,14,17,21)(15,24,23,38)(16,28,27,37)(18,32,31,22)(19,33,34,25)(20,29,36,30)(26,35,39,43)(40,41,44,42);(2,4)(13,15,17,23)(14,19,21,34)(16,26,27,39)(18,33,31,25)(20,35,36,43)(22,38,32,24)(28,40,37,44)(29,42,30,41);(5,8,9,7)(6,10,11,12)(15,25)(16,29)(18,31)(19,24)(20,37)(22,32)(23,33)(26,39)(27,30)(28,36)(34,38)(35,43);(13,16,17,27)(14,20,21,36)(15,26,23,39)(18,29,31,30)(19,35,34,43)(22,37,32,28)(24,40,38,44)(25,41,33,42)

Polycyclic presentation:

2^9|b^2=fi,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=dhi,h^c=hi,e^d=ehi,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, 6484059);