#131352

order	512
nilpotency class	4
isoclinism	isoclinic class of #131350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8403
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60771	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

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

normal subgroups only

This group has 753 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:

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

Polycyclic presentation:

2^9|a^2=f,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=eg,f^b=fgh,h^b=hi,d^c=dgi,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38548);