#120440

order	512
nilpotency class	3
isoclinism	isoclinic class of #10500
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#1441
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(4,5)(7,8)(9,10,14,18,15,20,13,19)(11,21,24,32,25,35,23,16)(12,26,30,36,29,17,28,34)(22,33,39,31,38,37,27,40);(1,2,5,7,6,8,4,3)(9,11,14,24,15,25,13,23)(10,16,19,35,20,32,18,21)(12,27,30,22,29,39,28,38)(17,33,36,40,26,37,34,31);(1,3,4,8,6,7,5,2)(9,12,13,28,15,29,14,30)(10,17,18,34,20,26,19,36)(11,22,23,27,25,38,24,39)(16,31,21,37,32,40,35,33)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fhi,e^c=ehi,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, 27636);