#131350

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

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

normal subgroups only

This group has 761 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,f^2=h,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, 38546);