#452670

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi

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, 359866);