#141888

order	512
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=ei,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49084);