#6344448

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	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

1632 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:

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

Polycyclic presentation:

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

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, 6251644);