#145967

order	512
nilpotency class	4
isoclinism	isoclinic class of #145967
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10599
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53163);