#142769

order	512
nilpotency class	4
isoclinism	isoclinic class of #142608
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
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

510 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,11,14,15)(4,9,5,8)(7,13,16,12)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,30,26,27);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,22,28)(18,23,20,26)(21,30,31,27)(24,32,29,25);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20,22,18)(19,26,28,23)(21,29,31,24)(25,30,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,e^c=egi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49965);