#7193177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
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

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,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=dhi,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, 7100373);