#7149417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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

1828 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)(9,10,14,21,15,19,23,20)(11,22,25,16,26,35,40,34)(12,27,29,33,30,36,28,17)(13,18,32,38,24,37,31,39);(2,5)(3,7)(9,11,23,40,15,26,14,25)(10,16,21,35,19,34,20,22)(12,24,29,31,30,13,28,32)(17,18,36,39,33,37,27,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,24)(13,26)(14,28)(15,30)(16,18)(19,33)(20,27)(21,36)(22,38)(23,29)(25,32)(31,40)(34,37)(35,39);(2,5)(6,8)(9,13,15,24)(10,18,19,37)(11,12,26,30)(14,32,23,31)(16,33,34,17)(20,39,21,38)(22,27,35,36)(25,29,40,28);(1,3,4,7)(2,6,5,8)(10,19)(16,34)(17,33)(18,37)(20,21)(22,35)(27,36)(38,39)

Polycyclic presentation:

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

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, 7056613);