#131438

order	512
nilpotency class	4
isoclinism	isoclinic class of #131436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8404
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60937	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

637 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339831
maximal normal subgroup of	#72339831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38634);