#146155

order	512
nilpotency class	4
isoclinism	isoclinic class of #143935
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

235 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dhi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53351);