#6353472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
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	C₄²	C₂⁵
prime radical	#63021	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2815 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:

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

Polycyclic presentation:

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

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, 6260668);