#7339801

order	512
nilpotency class	3
isoclinism	isoclinic class of #7339796
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#48088
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 7246997);