#135299

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42495);