#6439254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,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, 6346450);