#393645

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300841);