#7185432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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
prime radical	#3499	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7092628);