#7143225

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

1914 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)(9,10,15,19)(11,21,26,36)(12,27,13,18)(14,31,32,20)(16,25,35,24)(17,28,37,29)(22,34,30,33)(23,39,40,38);(2,5)(3,7)(9,11)(10,16)(12,22)(13,30)(14,25)(15,26)(17,33)(18,38)(19,35)(20,36)(21,31)(23,29)(24,32)(27,39)(28,40)(34,37);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,37)(11,22,26,30)(14,28,32,29)(16,33,35,34)(18,31,27,20)(21,39,36,38)(23,25,40,24);(2,5)(6,8)(9,13,15,12)(10,18,19,27)(11,23,26,40)(14,28,32,29)(16,34,35,33)(17,31,37,20)(21,39,36,38)(22,25,30,24);(1,3,4,7)(2,6,5,8)(10,19)(16,35)(17,37)(18,27)(20,31)(21,36)(33,34)(38,39)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050421);