#144595

order	512
nilpotency class	4
isoclinism	isoclinic class of #144595
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,9,11,15)(4,7)(5,8)(6,12,10,13)(14,16)(18,23,25,31)(19,27)(20,22)(24,32,26,28)(29,30);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,13)(14,16)(18,24,25,26)(23,28,31,32);(1,3)(2,6)(4,13)(5,11)(7,15)(8,10)(9,16)(12,14)(17,18)(19,28)(20,26)(21,25)(22,24)(23,29)(27,31)(30,32)

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=ghi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fg,h^a=hi,c^b=cf,f^b=fgi,d^c=dg,e^c=egh,f^c=fi,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51791);