#103309

order	512
nilpotency class	3
isoclinism	isoclinic class of #103301
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#28301	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

751 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 88 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)(11,13)(12,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,32)(29,31);(1,2,4,6)(3,8,7,5)(17,19)(18,22)(20,29)(21,27)(23,32)(24,31)(25,28)(26,30);(3,7)(5,8)(10,13)(11,16)(14,15)(22,31)(25,32)

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,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, 10505);