#7211754

order	512
nilpotency class	3
isoclinism	isoclinic class of #7211754
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#45599
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi,f^e=fi,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7118950);