#145641

order	512
nilpotency class	4
isoclinism	isoclinic class of #145621
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×Q₈	C₂³
prime radical	C₂³	#394
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

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

normal subgroups only

This group has 295 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,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=egi,f^c=fgi,h^c=hi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52837);