#6348011

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6255207);