#6439493

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

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,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, 6346689);