#6606644

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
prime radical	#61256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6513840);