#142069

order	512
nilpotency class	4
isoclinism	isoclinic class of #125297
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#329
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

289 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,18,14,19,27,28,12,16,20,32,25,17,33,34)(11,15,23,30,21,31,26,39,22,29,38,40,24,36,37,35);(2,5)(6,7)(9,11,20,37,25,21,27,23,12,22,13,26,14,24,33,38)(10,15,28,40,19,36,32,39,16,29,34,30,17,31,18,35);(1,3,4,8)(2,6,5,7)(10,16)(15,29)(17,19)(18,32)(28,34)(30,40)(31,36)(35,39)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=ei,d^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49265);