#145725

order	512
nilpotency class	4
isoclinism	isoclinic class of #145724
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,21)(11,22,25,35)(12,16,13,28)(14,31,27,20)(17,30,34,23)(18,29,19,26)(24,39,37,32)(33,38,36,40);(1,3)(2,5)(4,7)(6,8)(10,16)(11,23)(12,26)(13,29)(14,27)(18,20)(19,31)(21,28)(22,32)(24,40)(25,30)(33,36)(35,39)(37,38);(5,8)(9,11,12,23,27,37,29,40,15,25,13,30,14,24,26,38)(10,17,18,22,31,36,28,39,21,34,19,35,20,33,16,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52921);