#104196

order	512
nilpotency class	3
isoclinism	isoclinic class of #104181
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	#2856	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 582 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,d^c=dh,e^c=ei

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11392);