#139838

order	512
nilpotency class	4
isoclinism	isoclinic class of #139838
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 567 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)(3,11,14,5)(4,15,18,29)(6,9,21,23)(7,10)(8,17)(12,25,27,32)(13,20)(16,30)(19,26)(22,31)(24,28);(1,3,5,10)(2,7,14,11)(4,12,19,22)(6,13,24,9)(8,18,27,16)(15,17,26,25)(20,23,21,28)(29,31,30,32);(1,4)(2,8)(3,12)(5,19)(6,17)(7,18)(9,15)(10,22)(11,16)(13,26)(14,27)(20,32)(21,31)(23,29)(24,25)(28,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fgi,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

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47034);