#142127

order	512
nilpotency class	4
isoclinism	isoclinic class of #142125
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1987
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

306 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,8)(5,6)(9,10,14,17,13,21,31,18,15,20,32,37,30,19,12,29)(11,22,26,35,28,38,40,39,27,36,23,16,25,34,24,33);(1,3,7,2,5,8,4,6)(9,11,13,25,15,27,30,28)(10,16,21,39,20,35,19,33)(12,26,14,24,31,23,32,40)(17,36,18,38,37,22,29,34);(2,6)(3,8)(9,12,30,32,15,31,13,14)(10,17,21,18,20,37,19,29)(11,23,28,24,27,26,25,40)(16,22,39,34,35,36,33,38)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eg,d^2=gh,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fhi,h^b=hi

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, 49323);