#145691

order	512
nilpotency class	4
isoclinism	isoclinic class of #145658
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52887);