#120207

order	512
nilpotency class	3
isoclinism	isoclinic class of #120205
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fh,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27403);