#6803851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6803848
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#39148
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1115 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)(3,13)(4,15)(5,21)(6,10)(7,26)(8,12)(9,19)(11,22)(14,29)(16,20)(17,27)(18,24)(23,32)(25,28)(30,31);(1,3)(2,4,12,20)(5,14)(6,24)(7,15)(8,9)(10,17,28,29)(11,13,26,19)(16,22)(18,32,31,21)(23,27)(25,30);(1,4)(2,9)(3,12)(5,17)(6,18)(7,13)(8,20)(10,27)(11,16)(14,28)(15,26)(19,22)(21,30)(23,29)(24,32)(25,31);(1,5,22,25,8,23,7,6)(2,10,26,32,12,28,11,21)(3,14,16,30,9,27,15,24)(4,17,19,31,20,29,13,18);(1,6,7,23,8,25,22,5)(2,10,26,32,12,28,11,21)(3,14,16,30,9,27,15,24)(4,18,13,29,20,31,19,17)

Polycyclic presentation:

2^9|b^2=fgh,d^2=fi,e^2=fhi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,g^a=gi,h^a=hi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,g^c=gi

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, 6711047);