#139843

order	512
nilpotency class	4
isoclinism	isoclinic class of #139842
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9499
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	#3484	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

620 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238351, #72265901, #72266931, …
maximal normal subgroup of	#72238351, #72265901, #72266931, …
maximal quotient of	#72265902

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47039);