#134550

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

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

normal subgroups only

This group has 388 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)(11,19,23,32)(12,17)(13,25,26,16)(14,18)(15,21,28,33)(20,30)(22,36)(24,38,34,29)(27,35)(31,37)(39,40);(9,11,20,33,14,23,12,21)(10,15,17,19,18,28,30,32)(13,22,34,40,26,37,24,35)(16,27,29,31,25,39,38,36);(1,2,4,6)(3,8,7,5)(10,16,18,25)(13,26)(15,27,28,39)(17,29,30,38)(19,31,32,36)(22,37)(24,34)(35,40)

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,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, 41746);