#139889

order	512
nilpotency class	4
isoclinism	isoclinic class of #139706
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, prime radical	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 458 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) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fh,e^d=ei,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, 47085);