#7156204

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	#61085	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)(9,10,16,21,15,22,14,20)(11,13,19,17,25,31,24,30)(12,26,32,35,28,36,29,18)(23,37,34,27,38,33,39,40);(2,5)(3,7)(9,11)(10,17)(12,23)(13,21)(14,19)(15,25)(16,24)(18,33)(20,31)(22,30)(26,27)(28,38)(29,34)(32,39)(35,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,29,15,28,16,32)(10,18,21,26,22,35,20,36)(11,23,19,34,25,38,24,39)(13,27,30,37,31,40,17,33);(2,5)(6,8)(9,13,14,30,15,31,16,17)(10,19,20,11,22,24,21,25)(12,27,29,37,28,40,32,33)(18,34,36,23,35,39,26,38);(1,3,4,7)(2,6,5,8)(12,28)(18,35)(23,38)(26,36)(27,40)(29,32)(33,37)(34,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,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,e^c=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, 7063400);