#121682

order	512
nilpotency class	3
isoclinism	isoclinic class of #102079
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 482 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=i,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,d^c=di,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28878);