#128645

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×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	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	#121	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

315 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)(3,5)(4,7)(6,8)(9,10,14,19,16,18,33,37,15,20,32,25,13,29,31,39)(11,21,23,28,22,17,34,27,24,36,38,26,30,35,40,12);(2,5,7,6)(9,11)(10,17)(12,25)(13,30)(14,23)(15,24)(16,22)(18,36)(19,27)(20,35)(21,29)(26,37)(28,39)(31,40)(32,38)(33,34);(9,12,14,21,16,28,33,17,15,27,32,36,13,26,31,35)(10,11,19,23,18,22,37,34,20,24,25,38,29,30,39,40)

Polycyclic presentation:

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

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, 35841);