#7156217

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

1986 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)(9,10,13,17,14,19,21,18)(11,20,23,15,24,29,34,30)(12,16,27,25,26,31,35,32)(22,33,38,28,37,36,40,39);(2,5)(3,7)(9,10)(11,15)(12,25)(13,18)(14,19)(16,27)(17,21)(20,23)(22,36)(24,30)(26,32)(28,38)(29,34)(31,35)(33,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,21,34,14,24,13,23)(10,15,17,29,19,30,18,20)(12,22,35,40,26,37,27,38)(16,28,25,36,31,39,32,33);(2,5)(6,8)(9,12,21,35,14,26,13,27)(10,16,18,32,19,31,17,25)(11,22,34,40,24,37,23,38)(15,28,20,33,30,39,29,36);(1,3,4,7)(2,6,5,8)(12,26)(16,31)(22,37)(25,32)(27,35)(28,39)(33,36)(38,40)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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^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, 7063413);