#7156195

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	#61084	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1947 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)(9,10,14,19,15,21,31,20)(11,22,24,34,26,37,25,16)(12,27,28,17,29,35,13,18)(23,36,30,40,39,32,38,33);(2,5)(3,7)(9,11)(10,16)(12,23)(13,30)(14,25)(15,26)(17,32)(18,36)(19,37)(20,22)(21,34)(24,31)(27,33)(28,38)(29,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,28,15,29,31,13)(10,17,20,27,21,18,19,35)(11,23,25,38,26,39,24,30)(16,32,22,33,34,36,37,40);(2,5)(6,8)(9,13,31,29,15,28,14,12)(10,18,20,35,21,17,19,27)(11,23,25,38,26,39,24,30)(16,33,37,32,34,40,22,36);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=h,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, 7063391);