#6355476

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1573 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)(18,22)(19,25)(20,28)(24,30)(26,27)(29,32);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14)(17,18,20,22,21,24,28,30)(19,23,27,29,26,31,25,32);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,28)(30,32);(3,13)(4,7)(8,16)(11,14)(19,26)(23,31)(25,27)(29,32);(17,20,21,28)(18,22,24,30)(19,27,26,25)(23,29,31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,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, 6262672);