#144611

order	512
nilpotency class	4
isoclinism	isoclinic class of #144607
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2029
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	#3492	C₂²
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 487 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=di,b^2=d,e^2=gi,f^2=gi,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,f^b=fi,d^c=dg,e^c=egi,f^c=fgi,h^c=hi,f^e=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, 51807);