#145765

order	512
nilpotency class	4
isoclinism	isoclinic class of #145718
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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	C₂×D₄
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

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

normal subgroups only

This group has 391 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=d,d^2=g,e^2=g,f^2=g,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=fg,h^c=hi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52961);