#97701

order	512
nilpotency class	3
isoclinism	isoclinic class of #97690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

768 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,7,4,6)(5,8)(10,14,16,19)(15,23,24,22)(25,26,28,32,29,31,37,39)(27,33,35,40,34,30,36,38);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,19)(13,16)(14,21)(17,22)(18,24)(20,23)(25,27,29,34)(26,30,31,33)(28,36,37,35)(32,40,39,38);(1,4,5,3)(2,7,8,6)(9,11,13,18)(10,15,16,24)(12,20,21,17)(14,22,19,23)(26,31)(27,34)(28,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg

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, 4897);