#6438159

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2568 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^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, 6345355);