#121333

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=e,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,d^c=di,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28529);