#139884

order	512
nilpotency class	4
isoclinism	isoclinic class of #139702
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#120	C₂×C₄
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

553 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 553 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,f^c=fh,e^d=ei,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, 47080);