#7156205

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

1962 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)(9,10,13,17,14,19,25,18)(11,20,23,15,22,30,35,29)(12,16,26,24,27,32,38,31)(21,33,37,28,36,34,40,39);(2,5)(3,7)(9,10)(11,15)(12,24)(13,18)(14,19)(16,26)(17,25)(20,23)(21,34)(22,29)(27,31)(28,37)(30,35)(32,38)(33,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,23,14,22,25,35)(10,15,18,20,19,29,17,30)(12,21,26,37,27,36,38,40)(16,28,31,33,32,39,24,34);(2,5)(6,8)(9,12,25,38,14,27,13,26)(10,16,18,31,19,32,17,24)(11,21,35,40,22,36,23,37)(15,28,20,33,29,39,30,34);(1,3,4,7)(2,6,5,8)(11,22)(15,29)(20,30)(21,36)(23,35)(28,39)(33,34)(37,40)

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,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, 7063401);