#102676

order	512
nilpotency class	3
isoclinism	isoclinic class of #102649
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

609 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=g,e^2=i,f^2=g,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=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9872);