#121120

order	512
nilpotency class	3
isoclinism	isoclinic class of #121117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

379 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28316);