#129550

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3480	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36746);