#7182411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7089607);