#386198

order	512
nilpotency class	3
isoclinism	isoclinic class of #220210
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293394);