#7142822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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

1916 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)(9,10,16,22)(11,23,12,26)(13,29,32,19)(14,21,15,20)(17,28,18,25)(24,36,27,39)(30,34,40,35)(31,37,33,38);(2,5)(3,7)(9,11,16,12)(10,17,22,18)(13,30,32,40)(14,28,15,25)(19,36,29,39)(20,26,21,23)(24,33,27,31)(34,38,35,37);(1,2)(3,6)(4,5)(7,8)(9,12,16,11)(10,18,22,17)(13,27,32,24)(14,28,15,25)(19,35,29,34)(20,26,21,23)(30,31,40,33)(36,37,39,38);(2,5)(6,8)(9,13)(10,19)(11,24)(12,27)(14,33)(15,31)(16,32)(17,34)(18,35)(20,38)(21,37)(22,29)(23,39)(25,40)(26,36)(28,30);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050018);