#102854

order	512
nilpotency class	3
isoclinism	isoclinic class of #102785
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

515 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,d^c=dh,e^c=ei,f^c=fgi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050);