#6439753

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2055 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,20,22,34)(12,17,26,32)(14,28,15,19)(16,24,31,23)(21,35,36,40)(25,29,33,27)(30,38,39,37);(2,5)(6,8)(9,10)(11,20)(12,25)(13,18)(14,19)(15,28)(16,24)(17,27)(21,30)(22,34)(23,31)(26,33)(29,32)(35,37)(36,39)(38,40);(9,11,13,22)(10,16,18,31)(12,21,26,36)(14,24,15,23)(17,30,32,39)(19,34,28,20)(25,35,33,40)(27,38,29,37);(9,12)(10,17)(11,21)(13,26)(14,27)(15,29)(16,30)(18,32)(19,33)(20,35)(22,36)(23,37)(24,38)(25,28)(31,39)(34,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=fi,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, 6346949);