#141358

order	512
nilpotency class	4
isoclinism	isoclinic class of #141346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48554);