#120725

order	512
nilpotency class	3
isoclinism	isoclinic class of #10500
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

430 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27921);