#146132

order	512
nilpotency class	4
isoclinism	isoclinic class of #146132
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

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

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, 53328);