#507553

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=gh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=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, 414749);