#120203

order	512
nilpotency class	3
isoclinism	isoclinic class of #120201
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

454 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)(12,25,28,35)(13,19)(14,18)(15,20)(16,23)(17,27,36,30)(22,31,38,40)(24,32)(26,34,37,39)(29,33);(9,11,13,23,15,24,14,29)(10,16,18,21,20,33,19,32)(12,26,27,22,28,37,30,38)(17,34,35,40,36,39,25,31);(2,5)(3,7)(9,12,13,27,15,28,14,30)(10,17,18,35,20,36,19,25)(11,22,23,37,24,38,29,26)(16,31,21,34,33,40,32,39)

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,c^b=cf,f^b=fh,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27399);