#134529

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#392	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 431 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,f^b=fi,e^c=eh,f^c=fgh,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, 41725);