#139810

order	512
nilpotency class	4
isoclinism	isoclinic class of #139809
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9497
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	#3484	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 709 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)(3,13,16,26)(4,17,19,24)(5,11,6,10)(7,12)(8,15)(9,18)(14,28)(20,23)(21,25)(22,31)(27,29,32,30);(1,3,6,15,7,16,5,20)(2,8,11,26,12,23,10,13)(4,14,18,30,19,31,21,29)(9,22,17,32,25,28,24,27);(1,4)(2,9,12,25)(3,14)(5,21)(6,18)(7,19)(8,22,23,28)(10,24,11,17)(13,27,26,32)(15,30)(16,31)(20,29)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,f^c=fh,e^d=ei,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, 47006);