#6621378

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#471
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

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, 6528574);