#7156203

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

1957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,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, 7063399);