#7185428

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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)(9,10,16,22)(11,23,26,36)(12,27,29,37)(13,19,31,34)(14,21,15,20)(17,32,35,25)(18,24,38,28)(30,40,33,39);(1,2,5,6)(3,4,8,7)(9,11,16,26)(10,17,22,35)(12,13,29,31)(14,32,15,25)(18,34,38,19)(20,36,21,23)(24,33,28,30)(27,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,12,15,28,16,29,14,24)(10,18,21,27,22,38,20,37)(11,13,25,30,26,31,32,33)(17,34,23,40,35,19,36,39);(2,6)(3,8)(9,13)(10,19)(11,24)(12,25)(14,33)(15,30)(16,31)(17,27)(18,36)(20,40)(21,39)(22,34)(23,38)(26,28)(29,32)(35,37);(1,4,5,7)(2,3,6,8)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,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, 7092624);