#137532

order	512
nilpotency class	4
isoclinism	isoclinic class of #137497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=g,d^2=gi,e^2=gh,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44728);