#129598

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3460	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

614 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)(3,7,8,5)(4,6)(9,10,13,19,14,20,29,37,15,17,28,33,12,18,26,35)(11,21,24,38,23,31,39,36,22,34,27,40,25,16,30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,27)(14,23)(15,22)(17,31)(18,34)(19,36)(20,21)(24,28)(26,39)(29,30)(32,33)(35,38)(37,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,22)(13,28)(18,20)(23,25)(26,29)(32,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36794);