#7156220

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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	#61080	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1913 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)(9,10,14,20,16,22,15,21)(11,13,19,17,25,32,24,31)(12,26,28,36,30,35,29,18)(23,37,34,27,39,33,38,40);(2,5)(3,7)(9,11)(10,17)(12,23)(13,20)(14,24)(15,19)(16,25)(18,33)(21,32)(22,31)(26,27)(28,38)(29,34)(30,39)(35,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12,15,29,16,30,14,28)(10,18,20,26,22,36,21,35)(11,23,19,34,25,39,24,38)(13,27,31,37,32,40,17,33);(2,5)(6,8)(9,13,15,31,16,32,14,17)(10,19,21,11,22,24,20,25)(12,27,29,37,30,40,28,33)(18,34,35,23,36,38,26,39);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,22,21)(11,24,25,19)(12,28,30,29)(13,31,32,17)(18,26,36,35)(23,38,39,34)(27,37,40,33)

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,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063416);