#6380628

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287824);