#144649

order	512
nilpotency class	4
isoclinism	isoclinic class of #144648
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10392
commutator, Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

471 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=d,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268800
maximal normal subgroup of	#72268800
maximal quotient of	#72268860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51845);