#100565

order	512
nilpotency class	3
isoclinism	isoclinic class of #100555
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

456 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,d^2=gh,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7761);