#145679

order	512
nilpotency class	4
isoclinism	isoclinic class of #145662
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	#2541	C₂²
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

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

normal subgroups only

This group has 372 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=d,d^2=g,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,e^c=ei,f^c=fg,h^c=hi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52875);