#146001

order	512
nilpotency class	4
isoclinism	isoclinic class of #146001
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2065
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	Q₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

227 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=d,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,c^b=cf,f^b=fgi,d^c=dgi,e^c=eh,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, 53197);