#128448

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	#525	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,d^b=dg

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, 35644);