#6439469

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

2370 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)(15,21)(16,22)(17,23)(18,33)(19,24)(20,25)(26,36)(27,38)(28,39)(29,40)(30,42)(31,41)(32,44)(34,37)(35,43);(6,9)(10,12)(13,15,20,28)(14,21,25,39)(16,29,26,35)(17,30,32,31)(18,27,19,34)(22,40,36,43)(23,42,44,41)(24,37,33,38);(2,4)(13,16,20,26)(14,22,25,36)(15,17,28,32)(18,31,19,30)(21,23,39,44)(24,42,33,41)(27,35,34,29)(37,40,38,43);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,26)(16,28)(18,29)(19,35)(20,32)(21,36)(22,39)(24,43)(25,44)(27,31)(30,34)(33,40)(37,42)(38,41);(13,18,20,19)(14,24,25,33)(15,27,28,34)(16,30,26,31)(17,29,32,35)(21,37,39,38)(22,41,36,42)(23,43,44,40)

Polycyclic presentation:

2^9|a^2=f,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,d^b=dh,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, 6346665);