#6346163

order	512
nilpotency class	3
isoclinism	isoclinic class of #61559
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61432	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,f^b=fh,e^d=ehi

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253359);