#121611

order	512
nilpotency class	3
isoclinism	isoclinic class of #10326
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,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, 28807);