#144284

order	512
nilpotency class	4
isoclinism	isoclinic class of #144207
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×2₋¹⁺⁴	D₄
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

529 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,e^c=ei,f^c=fi,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, 51480);