#6355036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354981
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1588 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,e^c=eh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262232);