#104815

order	512
nilpotency class	3
isoclinism	isoclinic class of #104789
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

462 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dhi,c^2=i,e^2=gh,f^2=gi,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,f^b=fi,d^c=dg,e^c=egh,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12011);