#205543

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112739);