#104518

order	512
nilpotency class	3
isoclinism	isoclinic class of #104437
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

393 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

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, 11714);