#6438611

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

2100 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6345807);