#136441

order	512
nilpotency class	4
isoclinism	isoclinic class of #136369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

295 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=h,d^2=g,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ehi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43637);