#106587

order	512
nilpotency class	3
isoclinism	isoclinic class of #106570
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

891 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13783);