#6344450

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
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	C₂³×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1570 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)(3,10)(4,8)(5,15)(6,9)(7,14)(11,12)(13,16)(17,18,21,25,22,26,20,24)(19,27,30,31,28,32,29,23);(1,3,5,14,6,13,4,12)(2,7,8,10,9,11,15,16)(17,19,20,29,22,28,21,30)(18,23,25,27,26,31,24,32);(1,2)(3,11)(4,15)(5,8)(6,9)(7,13)(10,12)(14,16)(19,28)(23,31)(27,32)(29,30);(1,4,6,5)(2,8,9,15)(3,12,13,14)(7,10,11,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27);(3,13)(7,11)(10,16)(12,14)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251646);