#104139

order	512
nilpotency class	3
isoclinism	isoclinic class of #104053
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
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

708 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 96 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)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,28)(29,31)(30,32);(1,2,4,6)(3,8,7,5)(13,16)(18,23)(20,21)(24,27)(25,26)(28,30)(31,32);(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,24,26,32)(20,28,21,30)(23,31,25,27)

Polycyclic presentation:

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

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, 11335);