#100910

order	512
nilpotency class	3
isoclinism	isoclinic class of #100827
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

489 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=gh,d^2=gh,e^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eghi,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8106);