#101365

order	512
nilpotency class	3
isoclinism	isoclinic class of #101343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 473 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8561);