#6565851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535687
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	C₂²×C₄	C₂⁵
prime radical	#61321	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1681 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,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, 6473047);