#6439490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2363 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6346686);