#134519

order	512
nilpotency class	4
isoclinism	isoclinic class of #124850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1799
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ci,e^c=eh,f^c=fghi,f^e=fi

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, 41715);