#142125

order	512
nilpotency class	4
isoclinism	isoclinic class of #142125
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

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

270 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,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, 49321);