#354182

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ 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)(11,15)(12,16,22,28)(13,18)(14,17)(19,25,31,35)(20,27)(21,26)(23,29,34,38)(24,30)(32,36,39,40)(33,37);(2,5)(6,8)(9,11,13,20,14,21,24,33)(10,15,18,27,17,26,30,37)(12,19,23,32,22,31,34,39)(16,25,29,36,28,35,38,40);(9,12)(10,16,17,28)(11,19)(13,23)(14,22)(15,25,26,35)(18,29,30,38)(20,32)(21,31)(24,34)(27,36,37,40)(33,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,h^2=i,b^a=be,c^a=cf,f^a=fi,f^c=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261378);