#6439605

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^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,h^b=hi,g^c=gi,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, 6346801);