#134501

order	512
nilpotency class	4
isoclinism	isoclinic class of #124849
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×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	#392	D₄
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 86 normal subgroups are shown.

normal subgroups only

This group has 431 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,23)(13,26,27,16)(14,18)(15,20)(17,25)(21,35,36,31)(22,32)(24,30,38,37)(28,33,39,40)(29,34);(2,5)(6,7)(9,11,12,20,14,22,25,34)(10,15,17,19,18,29,23,32)(13,21,24,33,27,36,38,40)(16,28,30,31,26,39,37,35);(2,6)(4,8)(5,7)(10,16)(15,28)(17,30)(18,26)(19,31)(23,37)(29,39)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41697);