#142145

order	512
nilpotency class	4
isoclinism	isoclinic class of #125524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

255 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49341);