#104772

order	512
nilpotency class	3
isoclinism	isoclinic class of #104753
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

545 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dgh,c^2=hi,e^2=gh,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,f^b=fg,d^c=dg,e^c=egh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11968);