#134958

order	512
nilpotency class	4
isoclinism	isoclinic class of #134940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#461
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42154);