#103890

order	512
nilpotency class	3
isoclinism	isoclinic class of #103845
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

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

436 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 90 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)(10,15)(12,13)(16,23)(17,18)(19,24)(20,22)(25,26,30,28)(27,33,36,40)(29,37,39,32)(31,38,34,35);(9,10,14,17)(11,19,21,23)(12,18,13,15)(16,22,24,20)(25,27)(26,31)(28,34)(29,38)(30,36)(32,40)(33,37)(35,39);(2,5)(3,8)(6,7)(9,11,14,21)(10,16,17,24)(12,20,13,22)(15,19,18,23)(25,28,30,26)(27,34,36,31)(29,32,39,37)(33,38,40,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11086);