#102691

order	512
nilpotency class	3
isoclinism	isoclinic class of #102685
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

422 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,5)(4,7)(6,8)(10,14)(12,19)(15,23)(16,20)(17,21)(22,24)(25,26,29,32)(27,30,34,39)(28,35,36,31)(33,40,37,38);(3,8)(5,6)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23)(27,33)(28,36)(30,38)(31,35)(34,37)(39,40);(2,5)(3,8)(6,7)(9,11,13,18)(10,15,16,24)(12,21,19,17)(14,22,20,23)(25,27)(26,30)(28,37)(29,34)(31,40)(32,39)(33,36)(35,38)

Polycyclic presentation:

2^9|a^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=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9887);