#7156200

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
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	#61085	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063396);