#136381

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

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

normal subgroups only

This group has 303 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,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, 43577);