#104623

order	512
nilpotency class	3
isoclinism	isoclinic class of #104565
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

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

463 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

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, 11819);