#134194

order	512
nilpotency class	4
isoclinism	isoclinic class of #133908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2296 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 653 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41390);