#6344487

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1585 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:

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

Polycyclic presentation:

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

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, 6251683);