#104223

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	C₂⁴	#120
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 623 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,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, 11419);