#102734

order	512
nilpotency class	3
isoclinism	isoclinic class of #102709
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

477 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

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, 9930);