#6345887

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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	#61450	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253083);