#134551

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 94 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)(11,19,21,34)(12,17)(13,25,26,16)(14,18)(15,22,29,35)(20,31)(23,32)(24,39,36,30)(27,40)(28,38)(33,37);(9,11,20,35,14,21,12,22)(10,15,17,19,18,29,31,34)(13,23,36,40,26,37,24,38)(16,28,30,33,25,27,39,32);(1,2,4,6)(3,8,7,5)(10,16,18,25)(11,21)(13,26)(15,27,29,28)(17,30,31,39)(19,32,34,33)(22,35)(24,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41747);