#7148108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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	#60928	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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)(9,10,13,29,15,20,14,19)(11,21,30,35,25,34,24,16)(12,26,31,17,28,18,27,36)(22,33,39,38,40,32,23,37);(2,5)(3,7)(9,11)(10,16)(12,22)(13,30)(14,24)(15,25)(17,32)(18,37)(19,34)(20,35)(21,29)(23,27)(26,38)(28,40)(31,39)(33,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,22)(13,27)(14,31)(15,28)(16,32)(18,19)(20,36)(21,38)(23,30)(24,39)(25,40)(26,29)(33,35)(34,37);(2,5)(6,8)(9,12)(10,18)(11,23)(13,31)(14,27)(15,28)(16,33)(17,19)(20,26)(21,38)(22,30)(24,40)(25,39)(29,36)(32,35)(34,37);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7055304);