#6444166

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6351362);