#127792

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×D₄	C₂³
socle	C₂²	C₂×#342
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,4,7,5,8,3,6)(10,14)(15,21)(16,19)(22,23);(2,6,8,7)(9,10)(11,15)(12,19,20,14)(13,16)(17,23,24,21)(18,22);(9,11,13,18)(10,15,16,22)(12,17,20,24)(14,21,19,23)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=g,b^a=bd,d^a=dg,f^a=fgh,d^b=dh,h^b=hi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34988);